section 1.8: x () - kkuniyuk.com · (section 1.8: x = fy()) 1.8.3 • the graph of x = y 2 below is...

(Section 1.8: x = f y( ) ) 1.8.1

SECTION 1.8 : x = f y( )

LEARNING OBJECTIVES

• Know how to graph equations of the form x = f y( ) .

• Compare these graphs with graphs of equations of the form y = f x( ) .

• Recognize when a curve or an equation describes x as a function of y, and

apply the Horizontal Line Test (HLT) for this purpose.

• Know basic graphs in this new context.

• Adapt rules and techniques for function behavior, symmetry, and transformations

to this new context.

PART A: DISCUSSION

• Sometimes, x is treated as a function of y, and we graph equations of the form

x = f y( ) in the xy-plane. These graphs must pass the Horizontal Line Test (HLT).

• The ordered pairs we associate with f are now of the form output, input( ) .

• x and y switch roles. y, not x, is our independent variable, and x, not y, is our

dependent variable. Function values correspond to x values.

• Function evaluations and point-plotting are modified accordingly. The order of

the coordinates of ordered pairs must be kept in mind.

• We will investigate similarities and differences between graphs of equations of

the form x = f y( ) and those of the familiar form y = f x( ) . We will adapt tools

for drawing the new graphs.

• For example, we will consider intervals on which f is increasing or decreasing,

as well as intercepts. We will also manipulate equations to help us graph them.

• We will see how our rules and techniques for symmetry (from Sections 1.3 and

1.7) and transformations (from Section 1.4) need to be modified.

• We will develop a gallery of basic graphs similar to the one in Section 1.3.

• In Section 1.9, we will see that the graphs of x = f y( ) and y = f x( ) are

reflections about the line y = x .

(Section 1.8: x = f y( ) ) 1.8.2

PART B : GRAPHING x = f y( )

Graphing x = f y( )

As a set of ordered pairs, the graph of x = f y( ) is given by:

f y( ) , y( ) y Dom f( ){ } .

• The graph of x = f y( ) is typically different from the graph of y = f x( )

in the usual xy-plane.

• When we do function evaluations and point-plotting, we treat y as the

independent (“input”) variable and x as the dependent (“output”) variable.

• WARNING 1: We still order coordinates of points as

x, y( ) , but they now

take the form output, input( ) .

PART C: SQUARING FUNCTION and EVEN FUNCTIONS

Let f y( ) = y2

. f here is the same squaring function on we have always known.

• Its rule could also be given as: f x( ) = x2.

•

Dom f( ) = , and

Range f( ) = 0, ) , as before.

• A table for x = f y( ) is below.

Input

y

Output

x

f y( )

y2

Point

x, y( )

Input

y

Output

x

f y( )

y2

Point

x, y( )

0 0 0, 0( ) 0 0

0, 0( )

1 1 1,1( )

1 1

1, 1( )

2 4 4, 2( )

2 4

4, 2( )

3 9 9, 3( )

3 9

9, 3( )

(Section 1.8: x = f y( ) ) 1.8.3

• The graph of x = y2

below is a parabola opening to the right.

WARNING 2: The graph never goes to the left of the y-axis, because

squares of real numbers are never negative.

• Look at the table. f is even, so each pair of opposite y values yields

a common function value f y( ) , or x. Graphically, this means that every

point x, y( ) on the graph has a “mirror image partner” x, y( ) that is also

on the graph, and the graph is symmetric about the x-axis (WARNING 3).

A function f is even f y( ) = f y( ) , y Dom f( )

The graph of x = f y( ) is

symmetric about the x -axis.

• f is decreasing on the y-interval

, 0( , so x values decrease there, and

the graph moves to the left as we move up the graph on that interval.

• f is increasing on the y-interval

0, ) , so x values increase there, and

the graph moves to the right as we move up the graph on that interval.

TIP 1: We could try to solve the equation x = f y( ) for y in terms of x.

Here, x = y2 y = ± x . The graph of y = x is the top half of

the parabola, while the graph of y = x is the bottom half.

(Section 1.8: x = f y( ) ) 1.8.4

PART D: THE HORIZONTAL LINE TEST (HLT)

In Section 1.2, we said that an equation in x and y describes y as a function of x

its graph passes the Vertical Line Test (VLT) in the xy-plane. We now have:

The Horizontal Line Test (HLT)

A curve in a coordinate plane passes the Horizontal Line Test (HLT)

There is no horizontal line that intersects the curve more than once.

An equation in x and y describes x as a function of y

Its graph in the xy-plane passes the HLT.

• Then, there is no input y that yields more than one output x.

• Then, we can write x = f y( ) , where f is a function.

• A “curve” could be a straight line.

Observe that the graph of x = y2

from Part C passes the HLT.

PART E: ESTIMATING DOMAIN AND RANGE FROM A GRAPH

The domain of f is the set of all y-coordinates of points on the graph of

x = f y( ) . (Think of projecting the graph onto the y-axis.)

The range of f is the set of all x-coordinates of points on the graph of

x = f y( ) . (Think of projecting the graph onto the x-axis.)

WARNING 4:

Domain

Think: yf

Range

Think: x

• If f y( ) = y2

, then, once again, Dom f( ) = , and

Range f( ) =

0, ) .

The graph in Part D demonstrates this.

(Section 1.8: x = f y( ) ) 1.8.5

PART F: CUBING FUNCTION and ODD FUNCTIONS

Let f y( ) = y3

. Graph x = f y( ) .

• If we solve x = y3

for y, we obtain the equivalent equation y = x3

.

We saw the graph of y = x

3

in Section 1.3, and it is the graph that we want here.

• Observe that f is increasing on , meaning that the graph moves to the right as

we move up the graph.

WARNING 5: When graphing x = f y( ) , trace the graph from bottom-to-

top (in the direction of increasing y), not top-to-bottom. A common error is

to reflect the graph about the x-axis, which can happen if you rotate your

head clockwise and draw the shape of the graph of y = x3

.

• f is an odd function, and we have graphical symmetry about the origin,

just as before.

A function f is odd f y( ) = f y( ) , y Dom f( )

The graph of x = f y( ) is

symmetric about the origin.

(Section 1.8: x = f y( ) ) 1.8.6

PART G: TRANSLATIONS (“SHIFTS”) and INTERCEPTS

WARNING 6: This time, horizontal shifts are more intuitive than vertical shifts.

Let G be the graph of x = f y( ) .

Let c be a positive real number.

Horizontal Translations (“Shifts”)

The graph of x = f y( )+ c is G shifted right by c units.

• We are increasing the x-coordinates.

The graph of x = f y( ) c is G shifted left by c units.

Vertical Translations (“Shifts”)

The graph of x = f y c( ) is G shifted up by c units.

The graph of x = f y + c( ) is G shifted down by c units.

Example 1 (Translations)

Let f y( ) = y2

. G is the central, purple graph of x = f y( ) below.

§

(Section 1.8: x = f y( ) ) 1.8.7

The “coordinate shift method” from Section 1.4, Part F still applies:

Example 2 (Coordinate Shifting and Intercepts)

We will translate the parabola on the left so that its turning point (its vertex)

is moved from 0, 0( ) to

4,1( ) ; that is, it is moved 4 units to the left and

1 unit up.

Graph of x = y2

Graph of x + 4 = y 1( )2

, or

x = y 1( )

2

4

Look at the graph on the right. As usual,

• To find the x-intercept, substitute y = 0 into an equation, and solve for x.

• To find the y-intercepts, substitute x = 0 into an equation, and solve for y.

(Finding these intercepts algebraically will be an Exercise.) §

When finding intercepts in the x = f y( ) setting, there are some differences.

• There can be at most one x-intercept, corresponding to f 0( ) , if it exists.

• There can be any number of y-intercepts (possibly none), or infinitely

many, corresponding to the real zeros of f .

(Section 1.8: x = f y( ) ) 1.8.8

PART H: REFLECTIONS

Let G be the graph of x = f y( ) .

The graph of x = f y( ) is G reflected about the y-axis. (WARNING 7)

The graph of x = f y( ) is G reflected about the x-axis.

The graph of x = f y( ) is G reflected about the origin.

Example 3 (A Reflection about the y-axis)

The graph of x = y2

is in blue below. The graph of x = y2

is in red.

§

Example 4 (Reflections)

Let f y( ) = y . G is the upper right, purple graph of

x = f y( ) below.

Observe that x = y y = x2 x 0( ) , so G is the right half of an

upward-opening parabola.

§

(Section 1.8: x = f y( ) ) 1.8.9

PART I: NONRIGID TRANSFORMATIONS; STRETCHING AND SQUEEZING

WARNING 8: Just as for translations (“shifts”) for the x = f y( ) case, the

horizontal transformations are now the more intuitive ones.

The graph of x = cf y( ) is:

a horizontally stretched version of G if c > 1

a horizontally squeezed version of G if 0 < c < 1

The graph of x = f cy( ) is:

a vertically squeezed version of G if c > 1

a vertically stretched version of G if 0 < c < 1

If c < 0 , then perform the corresponding reflection either before or after the

horizontal or vertical stretching or squeezing.

Example 5 (Stretching and Squeezing)

Let f y( ) = y2. Consider x = cf y( ) on the left and x = f cy( ) on the right.

§

(Section 1.8: x = f y( ) ) 1.8.10

PART J: A GALLERY OF GRAPHS

• Most of the graphs below differ from those in the table in Section 1.3, Part P.

•• However, x = y3

is equivalent to y = x

3

, so they share a common graph.

(Which other graphs are familiar from Section 1.3?)

• Corresponding domains and ranges are the same as those in Section 1.3.

•• Again, they can be inferred from the graphs.

•• The domain corresponds to y-coordinates this time, the range to x.

• Graphs of even functions are now symmetric about the x-axis.

• We associate “increasing” with “move right” as we move up the graphs.

Equation (Sample)

Graph

Even/Odd;

Symmetry Equation

(Sample)

Graph

Even/Odd;

Symmetry

x = c

Even;

x-axis

x =1

y2

Even;

x-axis

x = y

Odd;

origin

x = y

Neither

x = cy + d

c 0( )

Odd

d = 0 ;

then,

origin

x = y3

Odd;

origin

x = y2

Even;

x-axis

x = y2/3

Even;

x-axis

x = y3

Odd;

origin x = y

Even;

x-axis

x=1

y

Odd;

origin

x = a2 y2

a > 0( )

Even;

x-axis

(Section 1.9: Inverses of One-to-One Functions) 1.9.1

SECTION 1.9: INVERSES OF ONE-TO-ONE FUNCTIONS

LEARNING OBJECTIVES

• Understand the purposes and properties of inverse functions.

• Recognize when a function is one-to-one and invertible, and apply the

Horizontal Line Test (HLT) for this purpose.

• Know how to find the inverse of a one-to-one function numerically and

graphically, and inverse formulas conceptually and mechanically.

PART A: DISCUSSION

• In Section 1.8, we saw that the graphs of x = y3

and y = x

3

were the same.

Each equation can be solved for the “other variable” to obtain the other equation.

We call the corresponding cubing and cube root functions a pair of inverse

functions. The graphs of y = x3

and y = x

3

are reflections about the line y = x .

• The term inverse is used in many different contexts. We discussed logical

inverses in Section 0.2. We will review additive and multiplicative inverses in

this section. Inverse functions are inverses with respect to composition of

functions (see Section 1.6). (Typically, f

11 / f .) Inverse properties are based

on the idea that a pair of inverse functions “undo” each other.

• Inverse functions are developed by switching inputs with outputs and thus

domains with ranges. In order for a function f to be invertible (to have an inverse

function f

1), f must be one-to-one so that we obtain a function after the switch.

Sometimes, the implied domain must be restricted for a function to be invertible.

• In Section 1.8, we used the Horizontal Line Test (HLT) to see if an equation or

its graph represented x as a function of y. Now, we will use the HLT to see if a

function is one-to-one and invertible.

• We use inverses to solve equations uniquely. For example, when solving

x +1= 3, we use subtraction by 1 to invert addition by 1.

• In Chapter 3, we will see that exponential and logarithmic functions form pairs of inverse

functions; in Chapter 4, we will discuss trigonometric and inverse trigonometric functions. We

will solve related equations using inverse properties. In Section 8.4, we will define inverse

matrices as multiplicative inverses, although they will be related to inverse functions

(“transformations”) in linear algebra.


PART B: INVERSE FUNCTIONS

We will begin by reviewing additive and multiplicative inverses.

• 0 is the additive identity of , because, if we add 0 to any real number, the

result is identical to that number. Similarly, 1 is the multiplicative identity.

• The additive inverse of, say, the real number 3 is

3. That is,

3 is the inverse

of 3 with respect to addition. This is because, if the two numbers are added, the

result is the additive identity, 0.

• The multiplicative inverse (or reciprocal) of 3 is 1

3. That is,

1

3 is the inverse of

3 with respect to multiplication. This is because, if the two numbers are

multiplied, the result is the multiplicative identity, 1.

A pair of inverse functions have the property that, when they are composed in

either order, the result is the identity function, which outputs its input. They are

inverses with respect to function composition. (See Footnote 1.)

If f has an inverse function (i.e., f is invertible), then this inverse is unique, and

it is denoted by f

1 (called f inverse).

• Also, the inverse of f

1 is f .

WARNING 1: f

1, the inverse function of f , is not typically the multiplicative

inverse of f . That is, typically,

f 11

f. The “

1” is just a superscript; it is not

an exponent. However,

f x( )1

is interpreted as

1

f x( ). (See Footnote 2.)

Inverse Properties (can be taken as the Definition of an Inverse Function)

f has a unique inverse function f

1

f 1 f x( )( ) = x ,

x Dom f( ) , and

f f 1 x( )( ) = x ,

x Dom f 1( ) .

That is, f and f

1 “undo” each other.


Example 1 (Temperature Conversion).

Let f be the function that converts temperature measures from the Celsius

scale to the Fahrenheit scale.

f1 then exists. It is the function that converts from Fahrenheit to Celsius.

• For example, f 0( ) = 32 and

f 1

32( ) = 0 , because 0 degrees

Celsius corresponds to 32 degrees Fahrenheit. That is, 0 C = 32 F .

Both measures give the freezing point of water at sea level.

• Also, f 100( ) = 212 and f 1212( ) = 100 , because

100 C = 212 F .

Both measures give the boiling point of water at sea level.

A partial table for f and f

1 is below.

f f

1

Input

x

Output f x( )

Input

x

Output

f 1 x( ) 0 32 32 0

100 212 212 100

A partial arrow diagram for f and f

1 is below.

The Inverse Properties are demonstrated below.

f 1 f 0( )( ) = f 1

32( ) = 0 f 1 f 100( )( ) = f 1

212( ) = 100

f f 132( )( ) = f 0( ) = 32 f f 1

212( )( ) = f 100( ) = 212

§


Example 1 demonstrates that, in going from a function to its inverse

(if it exists), inputs and outputs switch roles. This is a key theme.

“Input-Output” Properties of Inverse Functions

(can also be taken as the Definition of an Inverse Function)

If f has an inverse function f

1,

then f a( ) = b f 1 b( ) = a .

• That is, a, b( ) f b, a( ) f 1

.

Domain and Range of Inverse Functions

If f has an inverse function f

1, then:

Dom f( ) = Range f 1( ) , and

Dom f 1( ) = Range f( ) .

That is, the domain of one function is the range of the other.

Example 2 (Inverse Functions: Verification, Domain, and Range)

a) Let f x( ) = x3 on . Find

f1, and verify that it is the inverse of f .

b) Let g x( ) = x3

on

0, 2 . Find g

1.

c) Let h x( ) = x3

on

2,1,{ } . Find h1

.

§ Solution

The inverse of each cubing function above is a cube root function.

a) f 1 x( ) = x

3

on , which is Range f( ) . Let r x( ) = x

3

on .

We will verify that r is the inverse of f by verifying the Inverse

Properties for f and r.

r f( ) x( ) = r f x( )( ) = r x3( ) = x33

= x , x , and

f r( ) x( ) = f r x( )( ) = f x

3( ) = x3( )

3

= x , x .

b) g 1 x( ) = x

3

on

0, 8 , which is Range g( ) .

c) h 1 x( ) = x

3

on 8,1,3{ } , which is

Range h( ) .

The reader can investigate further in the Exercises. §


PART C: ONE-TO-ONE FUNCTIONS and

THE HORIZONTAL LINE TEST (HLT)

We now discuss which functions are invertible.

From Section 1.1,

A relation f is a function Each input to f in its domain yields

exactly one output in its range.

The graph of y = f x( ) in the xy-plane

passes the Vertical Line Test (VLT).

A function cannot allow an input in its domain to yield two or more

different outputs. The following do not represent functions:

Input-Output Machine Arrow Diagram Graph and Equation

Fails VLT

y = ± 9 x2

Now,

A function f is one-to-one Each output from f in its range is yielded by

exactly one input in its domain.

Equivalently, f a( ) = f c( ) a = c .

That is, identical outputs imply identical inputs.

(See Footnote 3.)

A one-to-one function cannot allow two or more different inputs (x values)

in its domain to yield the same output (y value).


Example 3 (A Function that is Not One-to-One: Unrestricted Squaring Function)

Let f x( ) = x2

on . Then, f is a function, but it is not one-to-one on , as

demonstrated by the figures below:

Input-Output Machine Partial Arrow Diagram Graph

Fails HLT (Section 1.8)

§

• Also, f a( ) = f c( ) a2

= c2 a = ± c , which is not equivalent to

a = c , if c 0 .

If we solve the equation f x( ) = 9 , or x

2= 9 , we obtain two solutions for x,

namely 3 and

3. These are the two answers to the question, “Whose square

is 9?”

• When we switch inputs with outputs, the expression f 1

9( ) is

not well-defined here, because there are two possible values it could

take on: 3 and

3.

As a consequence, f is not invertible, because it has no inverse function. §

However, a squaring function can be one-to-one and invertible on a restricted

domain, as we will see in Example 4.

• We will restrict domains when we define inverse trigonometric functions in Section

4.10.


Example 4 (A One-to-One Function: Squaring Function on a Restricted Domain;

Modifying Example 3)

Let g x( ) = x2

on the restricted domain

0, ) .

The graph of y = g x( ) below passes the Vertical Line Test (VLT) and also

the Horizontal Line Test (HLT).

Consequently, g is a one-to-one correspondence between Dom g( ) ,

the set of input x values, and Range g( ) , the set of output y values.

(Think of matched pairs.) Also, g a( ) = g c( ) a2

= c2 a = c ,

since only nonnegative inputs are allowed.

• If we solve the equation g x( ) = 9 , or

x2

= 9 x 0( ) , we obtain a

unique solution for x, namely 3. It is the unique input that yields 9.

On the graph of g above, the only point with y-coordinate 9 has

x-coordinate 3.

• More generally, g x( ) = b has a unique solution for x, the unique

input that yields b, whenever b is in Range g( ) , which is 0, ) .

We can define a unique inverse function g

1.

• Let Dom g 1( ) = Range g( ) , which is 0, ) .

• Define g 1 b( ) to be the unique solution to

g x( ) = b ,

for every b in Dom g 1( ) . For instance, g 1

9( ) = 3. (In Example 1,

since f 0( ) = 32 , we reverse the arrow and define f 132( ) to be 0.) §


In summary …

A function f is invertible

f x( ) = b has a unique solution, given by

x = f 1 b( ) ,

b Range f( )

f is one-to-one

The graph of y = f x( ) in the xy-plane passes the Horizontal Line Test (HLT).

• The one-to-one property is essential for a function to be invertible,

because we need the inverse to be a function after inputs are switched with

outputs.

• (See Footnote 3; we assume the “onto” property.)

PART D: GRAPHING INVERSE FUNCTIONS

By the “Input-Output” Properties, a, b( ) f b, a( ) f 1

.

Graphical Properties of Inverse Functions

If f is invertible, then:

The point a,b( ) lies on the graph of f

The point b,a( ) lies on the graph of f

1.

To obtain the graph of f

1, reflect the graph of f

about the line y = x .

Example 5 (Restricted Squaring Function; Revisiting Example 4)

Again, let g x( ) = x2

on

0, ) .

Partial tables for g and g

1 can be constructed as follows:

g g

1

x g x( ) Point

x g 1 x( ) Point

0 0 0, 0( ) 0 0 0, 0( )

1 1 1,1( ) 1 1

1,1( )

2 4 2, 4( ) 4 2

4, 2( )

3 9 3, 9( ) 9 3

9, 3( )


The graph of g, or y = g x( ) , is in blue below. It passes the HLT, meaning

that g is one-to-one.

Therefore, when we reflect the graph about the line y = x (drawn as a

dashed line, though it is not an asymptote), we obtain the red graph of

y = g 1 x( ) below, and it passes the VLT. It should look familiar …..

• The graph of y = g 1 x( ) is the graph of y = x , because

g 1 x( ) = x . It

makes sense that the corresponding square root function undoes what the

(restricted) squaring function does.

• It is also the graph of x = g y( ) , or x = y2

, with y restricted to 0, ) .

This is consistent with the aforementioned Graphical Properties of Inverse

Functions.

The graph of x = g y( ) for a function g is the reflection of the graph

of y = g x( ) about the line y = x .

• This holds even if g is not one-to-one. §


PART E: FINDING FORMULAS FOR INVERSE FUNCTIONS

If f is a one-to-one function, and if its formula can be expressed algebraically,

then we should be able to find a formula for f

1.

Conceptual Approach to Finding the Inverse of a One-to-One Function f

To determine f

1, we need to invert (or “undo”) the steps applied by f

in reverse order (WARNING 2). (See the Exercises.)

• Also, if necessary, impose restrictions and ensure that


Dom f 1( ) = Range f( ) . (See Example 10.)

TIP 1: Similarly, when dressing, you put on your socks before your shoes,

but, when undressing, you remove your shoes before your socks.

Example 6 (Conceptual Approach to Finding an Inverse)

Let

f x( ) =

x3

+ 5

7 on . Find

f 1 x( ) .

§ Solution

• What does f do to x?

1) Takes the cube root: “

input3

”

2) Adds 5 to the result from 1): “ input +5”

3) Divides the result from 2) by 7: “

input

7”

• What should f

1 do to its input? (The asterisk “*” denotes inverting.)

3*) Multiplies by 7: “ input 7 ”

2*) Subtracts 5 from the result from 3*): “ input 5”

1*) Cubes the result from 2*): “ input( )

3

”

• Therefore, f 1 x( ) = 7x 5( )

3

.

• Dom f( ) = Range f 1( ) = , and Dom f 1( ) = Range f( ) = , so no

restrictions are necessary. §


Mechanical Approach to Finding the Inverse of a One-to-One Function f

Given a formula for f in terms of x, we can attempt to find a formula for

f1 as follows:

Step 1: Replace f x( ) with y.

Step 2: Switch x and y.

(Remember the theme of switching inputs with outputs.)

Step 3: Solve for y, if possible.

Step 4: Replace y with f 1 x( ) .

Step 5: If necessary, impose restrictions and ensure that

Dom f( ) = Range f 1( ) , and Dom f 1( ) = Range f( ) .

• In Steps 1-3, we are essentially solving x = f y( ) for y.

Example 7 (Mechanical Approach to Finding an Inverse; Revisiting Example 6)

Again, let

f x( ) =

x3

+ 5

7 on . Find

f 1 x( ) .

§ Solution

Step 1: Replace f x( ) with y.

y =

x3

+ 5

7

Step 2: Switch x and y.

(Remember the theme of switching inputs with outputs.)

x =

y3+ 5

7


Step 3: Solve for y.

7x = y3 + 5 Multiplied both sides by 7( )

7x 5 = y3Subtracted 5 from both sides( )

7x 5( )3

= y Cubed both sides( )

WARNING 3 : Squaring both sides of an equation may or

may not yield an equivalent equation.

y = 7x 5( )3

• Compare these steps with Steps 3*, 2*, and 1* in Example 6.

Step 4: Replace y with f 1 x( ) , if we obtain a function.

f 1 x( ) = 7x 5( )

3

• f

1 is a function, because f was one-to-one.

Step 5: If necessary, impose restrictions and ensure that


Dom f 1( ) = Range f( ) .

• Dom f( ) = Range f 1( ) = , and

Dom f 1( ) = Range f( ) = , so

no restrictions are necessary. §

Example 8 (Evaluating an Inverse Function; Revisiting Examples 6 and 7)

Again, let f x( ) =x

3

+ 5

7 on . Evaluate f 1

1( ) .

§ Solution

f 1 x( ) = 7x 5( )3

on from Examples 6 and 7( )

f 11( ) = 7 1( ) 5( )

3

= 8

Absent Examples 6 and 7, we could have also solved f x( ) = 1 for x. §


Example 9 (Checking an Inverse Function Formula; Revisiting Examples 6 and 7)

Again, let f x( ) =x

3

+ 5

7 on . Let g x( ) = 7x 5( )

3

on .

Check that g = f 1

.

§ Solution

We will verify that the compositions g f and f g are identity functions.

Either check below is sufficient, because f is one-to-one, and

Range f( ) = Dom g( ) (each set is ). If in doubt, do both checks.

(See Footnote 1.)

Check that

g f( ) x( ) = x , x . Check that

f g( ) x( ) = x , x .

x , x ,

g f( ) x( ) = g f x( )( )

= gx

3

+ 5

7

= 7x

3

+ 5

75

3

= x3

+ 5 5

3

= x3

3

= x

f g( ) x( ) = f g x( )( )

= f 7x 5( )3( )

=7x 5( )

33 + 5

7

=7x 5+ 5

7

=7x

7

= x

Therefore, g = f 1

. §


Example 10 (Temperature Conversion; Revisiting Example 1)

Let f be the one-to-one function that converts from Celsius to Fahrenheit.

Then, f

1 is the function that converts from Fahrenheit to Celsius.

The following is left for the reader in the Exercises:

• Show that f x( ) =9

5x + 32 by developing a linear model for f such

that f 0( ) = 32 and f 100( ) = 212 .

• Show that f 1 x( ) =5

9x 32( ) in three different ways:

•• Develop a linear model for f

1 such that

f 1

32( ) = 0 and

f 1212( ) = 100 .

•• Begin with

f x( ) =

9

5x + 32 and apply the Conceptual

Approach used in Example 6.

•• Begin with

f x( ) =

9

5x + 32 and apply the Mechanical

Approach used in Example 7.

We will now determine domains and ranges.

(See Step 5 in the Mechanical Approach.)

• Temperatures cannot go below absolute zero, which is 273.15 C .

We require: Dom f( ) = Range f 1( ) = x x 273.15{ } .

• Find the Fahrenheit equivalent of 273.15 C .

f x( ) =9

5x + 32

f 273.15( ) =9

5273.15( ) + 32

= 459.67 F( )

• We require: Dom f 1( ) = Range f( ) = x x 459.67{ } .


Observe that the red graph for f and the brown graph for f

1 below are

reflections about the line y = x .

FOOTNOTES

1. Identity functions and compositions of inverse functions. There are technically different

identity functions on different domains.

(See Footnote 3 below and Section 1.1, Footnote 1.)

• Let f be an invertible function that maps from domain X to codomain Y ; i.e., f : X Y .

If f is invertible, then f is onto, meaning that the range of f is the codomain Y.

f1 maps from Y to X ; i.e.,

f1:Y X .

• Let I

X be the identity function on Dom f( ) , which is X .

I

X: X X .

• Let IY

be the identity function on Dom f 1( ) , which is Y . I

Y:Y Y .

• Then, f 1 f = I

X, and

f f 1

= IY

.

• If g is a function such that g f = I

X, then g is a left inverse of f ; f has a left inverse

f is one-to-one. For example, let X = 1, 2{ } ,

Y = 10, 20, 30{ } ,

f = 1,10( ), 2, 20( ){ } , and

g = 10,1( ), 20, 2( ), 30, 2( ){ } . Then, g is a left inverse of f .

• If h is a function such that f h = I

Y, then h is a right inverse of f ; f has a right inverse

f is onto. For example, let X = 1, 2, 3{ } ,

Y = 10, 20{ } ,

f = 1,10( ), 2, 20( ), 3, 20( ){ } ,

and h = 10,1( ), 20, 2( ){ } . Then, h is a right inverse of f , although h is not a left inverse of f .

• If f is one-to-one and onto, then f has a unique inverse function that serves as both a

unique left inverse and a unique right inverse.


2. fn. Many instructors reluctantly use the f 1

notation to represent the inverse function of f .

• This is because n often represents an exponent in the notation fn, except when

n = 1 .

For example, f 2 is often taken to mean ff ; that is,

f 2 x( ) = f x( ) f x( ) .

In Chapters 4 and 5, we will accept that sin

2 x = sin x( ) sin x( ) , which is the standard

interpretation.

• On the other hand (and this compounds the confusion), some sources use n to indicate the

number of applications of f in compositions of f with itself; the result is called an iterated

function. For example, they would let f 2= f f , and they would use the rule:

f 2 x( ) = f f x( )( ) . This is typically different from the rule

f 2 x( ) = f x( ) f x( ) .

However, our use of the notation f

1 for “ f inverse” is more consistent with this second

interpretation, since f

1 f is an identity function, which could be construed as f

0 in this

context.

3. One-to-one, onto, and bijective functions. A function f is invertible it is one-to-one

(or injective) and onto (or surjective); then, f is a bijective function, or a one-to-one

correspondence.

• A one-to-one (or “injective”) function has the property that, whenever f a( ) = f c( ) for

domain elements a and c, it must be true that a = c . That is, two outputs are equal the

inputs are equal. This definition will lead to the One-to-One Properties for exponential and

logarithmic functions in Chapter 3.

• An onto (or “surjective”) function has the property that its range is the entire codomain

(see Section 1.1, Footnote 1). This means that every element of the codomain is the image

(that is, function value or output) of some element of the domain.

• When we develop the inverse of a bijective function, we switch inputs with outputs.

•• The one-to-one property guarantees that the resulting function does not allow one

input to yield more than one output.

•• The onto property guarantees that the resulting function is defined for all elements of

the codomain of the original function, which is now the domain of the inverse function.

The onto property is often ignored in discussions of inverse functions. This is because we

typically force the codomain of the original function to be the range (or “image”) of the

original function in this context; this then guarantees the onto property. (Precalculus

sources typically avoid the term “codomain” in the first place.) If this is the case, then

“one-to-one function” and “one-to-one correspondence” are interchangeable, and we only

have to worry about the one-to-one property when it comes to invertibility.

section 1.8: x () - kkuniyuk.com · (section 1.8: x = fy()) 1.8.3 • the graph of x = y 2 below is...

Documents