functions pure maths

8/12/2019 Functions Pure Maths

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Functions

1) Transformations

Let us start with a function, in this case it

is f(x) = x2, but it could be anything:

f(x) = x2

Here are some simple things we can do to

move or scale it on the graph:

We can move it up or down by

adding a constant to the y-value:

g(x) = x2+ C

Note: to move the line down, we use

a negativevalue for C.

C > 0 moves it up C < 0 moves it down

We can move it left or right by

adding a constant to the x-value:

g(x) = (x+C)2

Adding Cmoves the function to

the left(the negative direction).

Why?Well imagine you will inherit a

fortune when your age=25. If you

change that to(age+4) = 25then you

would get it when you are 21. Adding 4

made it happen earlier.

C > 0 moves it left

C < 0 moves it right

An easy way to remember what happens

to the graph when we add a constant:

add to y: go high

add to x: go left

BUT we must add C wherever x

appearsin the function (we

are substituting x+C for x).
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Example: the function v(x) = x3-

x2+ 4x

Move C spaces to the left: w(x) =

(x+C)3- (x+C)2+ 4(x+C)

We can stretch or compress it in

the y-direction by multiplying the

whole function by a constant.

g(x) = 0.35(x2)

C > 1 stretches it

0 < C < 1 compresses it

We can stretch or compress it in

the x-direction by multiplying x(wherever it appears) by a

constant.

g(x) = (2x)2

C > 1 compresses it

0 < C < 1 stretches it

Note that (unlike for the y-

direction), biggervalues cause

more compression.

We can flip it upside down by

multiplying the whole function by -

1:

g(x) = -(x2)

This is also called reflection about the

x-axis(the axis where y=0)

We can combine a negative value with a

scaling.

Example: multiplying by -2 will flip itupside down AND stretch it in the y-

direction.

We can flip it left-right by

multiplying the x-value by -1:

g(x) = (-x)2


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It really does flip it left and right!But

you can't see it,

because x2is symmetrical about the y-

axis .So here is another example

using (x):

g(x) = (-x)

This is also called reflection about the

y-axis(the axis where x=0)

Summary

y = f(x) + C C > 0 moves it up

C < 0 moves it down

y = f(x + C) C > 0 moves it left

C < 0 moves it right

y = Cf(x) C > 1 stretches it in the y-direction

0 < C < 1 compresses it

y = f(Cx) C > 1 compresses it in the x-direction

0 < C < 1 stretches it

y = -f(x) Reflects it about x-axis

y = f(-x) Reflects it about y-axis

Example: the function g(x) = 1/x

Move 2 spaces up: h(x) = 1/x + 2

Move 3 spaces down: h(x) = 1/x - 3

Move 4 spaces to the right: h(x) =

1/(x-4)

Move 5 spaces to the left: h(x) =

1/(x+5)

Stretch it by 2 in the y-direction:

h(x) = 2/x

Compress it by 3 in the x-direction:

h(x) = 1/(3x)

Flip it upside down:

h(x) = -1/x
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Example: the function v(x) = x3- 4x

Move 2 spaces up: w(x) = x3- 4x + 2

Move 3 spaces down: w(x) = x3- 4x - 3

Move 4 spaces to the right: w(x) = (x-4)3- 4(x-4)

Move 5 spaces to the left: w(x) = (x+5)3- 4(x+5)

Stretch it by 2 in the y-direction: w(x) = 2(x3- 4x) = 2x3- 8x

Compress it by 3 in the x-direction: w(x) = (3x)3- 4(3x) = 27x3- 12x

Flip it upside down: w(x) = -x3+ 4x

do all transformation in one gousing this:

ais vertical stretch/compression

|a| > 1 stretches |a| < 1 compresses

a < 0 flips the graph upside down

bis horizontal stretch/compression

|b| > 1 compresses

|b| < 1 stretches

b < 0 flips the graph left-right

cis horizontal shift

c < 0 shifts to the right

c > 0 shifts to the left

dis vertical shift

d > 0 shifts upward

d < 0 shifts downward

Example: 2(x+1)+1

a=2, c=1, d=1

So it takes the square root function, and

then

Stretches it by 2 in the y-

direction

Shifts it left 1, and

Shifts it up 1


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2) Mapping(or function)

This a 'notation' for expressing a relation between

two variables(sayxandy).

Individual values of these variables are

called elements

eg x1x2x3... y1y2y3...

The first set of elements (x) is called the domain.

The second set of elements (y) is called the range .

A simple relation likey=x2can be more accuratelyexpressed using the following format:

The last part relates to the fact thatxandyareelements of the set of real numbers R(any positive or

negative number, whole or otherwise, including zero)

One-One mapping

Here one element of the domain is associated with

one and only one element of the range.

A property of one-one functions is that a on a graph

a horizontal line will only cut the graph once.

Example

R+the set of positive real numbers

Many-One mapping

Here more than one element of the domain can be

associated with one particular element of the range.

Example

Z is the set of integers(positive & negative whole

numbers not including zero)

Completefunction notation is a variation on what

has been used so far. It will be used from now on.


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Inverse Function f -1

The inverse functionis obtained by

interchanging xandyin the function equation and

then rearranging to makey

the subject.

If f -1exists then,

ff-1(x) = f-1f(x) =x

It is also a condition that the two functions be 'one to

one'. That is that the domain of fis identical to the

range of its inverse function f -1.

When graphed, the function and its inverse are

reflections either side of the line y=x.

Example

Find the inverse of the function(below) and graph thefunction and its inverse on the same axes.

Composite Functions

A composite functionis formed when two

functions f, gare combined.

However it must be emphasized that the order in

which the composite function is determined is

important.

The method for finding composite functions is:

findg(x)

findf[g(x)]

Example : For the two functions,

find the composite functions (i fg (ii g f


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Exponential & Logarithmic Functions

Exponential functions have the general form:

where 'a' is a positive constant

However there is a specific value of 'a' at (0.1) when

the gradient is 1 . This value, 2.718...or 'e' is called

theexponential function.

The function(above) has one-one mapping. It

therefore possesses an inverse. This inverse is

the logarithmic function.

1. Which of the following represent a mapping?

(a) {(4, 2); (5, 3); (7, 5); (9, 7)}(b) {(2, 8); (3, 12); (4, 16)}

(c) {(3, 7); (3, 11); (4, 9); (5, 11)}(d) {(1, 2); (2, 3); (3, 4); (4, 5)}

(e) {(2, 1); (3, 1); (5, 1); (7, 1)}

(f) {(1, 3); (1, 5); (2, 5)}

2. Which of the following arrow diagrams represent a mapping?

Give reasons.


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3. A function f is defined by f(x) = 2x - 3.Write the values of

(a) f(0)(b) f(-2)

(c) f(3)

(d) f(-1)

4. Find the domain and range of each of

the following functions.

(a) f(x) = 2 - x, x N

(b) f(x) = x2+ 1, x W

(c) f(x) = x, x R

5. Let A = {1, 3, 5, 7) and B = {3, 5, 7, 9

11}Consider the rule f(x) = x + 2, where x A.

Represent the mapping in the roster form.Also, find the domain and range of the

mapping.

6. Let A = {1, 2, 3} B = {3, 6, 9, 12, 15}Draw the arrow diagram to represent the rule

f(x) = 3x from A to B.

7. Let A = {3, 8, 11} and B = {1, 2, 3}

(a) Show that the relation R = {(3, 1), (8, 2)}

is not a mapping from A to B.

(b) Show that the relation R = {(3, 1); (3, 3);

(8, 2); (11, 1); (11, 3)} from A to B is not amapping from A to B.

8. Let A = {2, 3, 4} and B = {5, 9, 13}

Consider the rule f(x) = 4x - 3, where x A

(a) Show that f is a mapping from A to B.

(b) Find the domain and range of the mapping.(c) Represent the mapping in the roster form.

(d) Draw the arrow diagram to represent themapping.

Answers:

1. (a), (b), (d), (e)

2. (a) Since, every element of the domain has a

unique image in the co-domain.

3. (a) -3

(b) -7(c) 3(d) -5

4. (a) domain N Range= {1, 0, -1, -2...}

(b) Domain W Range = {1, 2, 5, 10, 17...}(c) domain R Range R

5. F = {(1, 3) (3, 5) (5, 7) (7, 9)} Domain =

{1, 3, 5, 7} Range = {3, 5, 7, 9}

6.

7. (a) domain {3, 8} A hence not a mapping(b) Elements 3, 11 do not have unique image in

B hence not a mapping

8. Ordered pairs {(2, 5), (3, 9), (4, 13)}

Elements of A have unique image in B hence amapping

Domain {2, 3, 4} Range {5, 9, 13}

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4) Even and Odd functionsEven Functions

A function is "even" when:

f(x) = f(-x) for all x

In other words there is symmetry about

the y-axis (like a reflection):

This is the curve f(x) = x2+1

They got called "even" functions because

the functions x2, x4, x6, x8, etc behave

like that, but there are other functions

that behave like that too, such as cos(x):

Cosine function: f(x) = cos(x)

It is an even function

But an even exponent does not always make an even function, for

example (x+1)2is notan even function.

Cosine curve:

f(t) = 2 cos t

Notice that we have a mirror image

through the f(t)axis.

Even Square wave

Triangular wave

In each case, there isa mirror image

through the f(t)axis.

- havesymmetryabout thevertical axi

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Odd Functions

A function is "odd" when:

-f(x) = f(-x) for all x

Note the minus in front of f: -f(x).

And we get origin symmetry :

This is the curve f(x) = x3

-x

They got called "odd" because the

functions x, x3, x5, x7, etc behave like

that, but there are other functions that

behave like that, too, such assin(x):

Sine function: f(x) = sin(x)

It is an odd function

But an odd exponent does not always

make an odd function, for

example x3+1is notan odd function.

Origin Symmetry

A graph has origin symmetryif we can fold it along

the verticalaxis, then along the horizontalaxis, and itlays the graph onto itself.

.Sine Curve: y(x) = sinx

Notice that if we fold the curve along

they-axis, then along the t-axis, thegraph maps onto itself. It has originsymmetry.

Types of waves:

Each of these three curves is an oddfunction, and the graphdemonstrates symmetry about theorigin
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Neither Odd nor Even

Don't be misled by the names "odd" and "even"

... they are just names... and a function

does not have to beeven or odd.

In fact most functions are neither odd nor even.

For example, just adding 1 to the curve above

gets this:

This is the curve f(x) = x3-x+1

It is not an odd function, and it is not an

even functioneither.

It is neither odd nor even!

Even or Odd?

Example: is f(x) = x/(x2-1) Even or

Odd or neither?

Let's see what happens when we

substitute -x:

Put in "-x": f(-x)= (-x)/((-x)2-1)

Simplify: = -x/(x2-1)

= -f(x)

So f(-x) = -f(x)and hence it is an Odd

Function

Special Properties

Adding:

The sum of two even functions is even

The sum of two odd functions is odd

The sum of an even and odd function is

neither even nor odd (unless one

function is zero).

Multiplying:

The product of two even functions is an

even function.

The product of two odd functions is an

even function.

The product of an even function and anodd function is an odd function.


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5) Composite Functions

If you have 3 sets, A, B and C, and f(x) maps Ato B, and g(x) maps B to C, then the composite

function that maps set A to B is gf(x) (that is,you 'plug' f(x) into g(x)). The diagram above

can also be redrawn to map A directly to C as

follows:

If you plug a function into itself, e.g. f[f(x)], itcan be written as f(x). If you plug f(x) into

itself twice, e.g. f[f(f(x))] = f[f(x)] = f(x).

If you're given 2 functions, g(x) andf(x), the order of mapping is important,

e.g. gf(x) fg(x), in most cases.However, there are exceptions.

Example 1 Functions where orderis importantThe functions f and g are defined

by f(x) = 2x + 1 and g(x) = x 2respectively. Find the functions (a) fg,

(b) gf

Solution(a)fg(x)

= f[g(x)]= f[x 2]= 2(x 2) + 1= 2x 3

(b) gf(x)= g[f(x)]= g[2x + 1]= (2x + 1) 2= 4x + 4x +1 2= 4x + 4x 1

Example 2 Functions where orderis not important

(a) fg(x) (b) gf(x)

Solution(a)

(b)

Example 3 Repeated functioncompositionGiven the following function,

Find the expression for each of the

following functions.

Solution

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Example 4 Finding a function

when given another function and acomposed function

(a) The function fis defined byf(x) = 2x + 4. Another function gissuch that fg(x) = 3x 8. Find thefunction g.(b) The function fis defined byf(x) = 1 x. Another function gis such

that gf(x) = 3x 6x + 5. Find thefunction g.

Solution

If you re-write uas x, you'll see thatg(x) = 3x + 2.

Example 5 Further examples oncomposite functionsGiven the functions f(x) = hk + k,g(x) = (x + 1) + 2 and

fg(x) = 2(x + 1) + 1, find(a) the value of g(2),

(b) the value of h and of k.

Solution(a)





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(b)It is given that f(x) = hx + k and g(x) =

(x + 1) + 2Thus,

But it is given that fg(x) = 2(x + 1) +1. Hence, by comparison,h = 2and

Example 6 Another furtherexample

Given the functions f(x) = 3x + 7 andfg(x) = 22 3x, find gf(x).

Solution

Find g(x) first,

Hence,

Question 1:

For f(x) = 2x + 3 and g(x) = -x2+ 1, find

the composite function defined by(f og)(x)

Solution to Question 1:

The definition of compositefunctions gives

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

= 2 (g(x)) + 3

= 2( -x2+ 1 ) + 3

= - 2 x2+ 5

Question 2:

Given f(2) = 3, g(3) = 2, f(3) = 4 and g(2)= 5, evaluate

(f og)(3)Solution to Question 2:

Use the definition of the compositefunction to write

(f og)(3) = f(g(3))

Substitute g(3) by its given value 2and evaluate f(2)

(f og)(3) = f(2) = 3

Question 4:

Functions f and g are as sets of orderedpairs

f = {(-2,1),(0,3),(4,5)}






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Solution to Question 10:

We first find h(1)

h(1) = - 1

We now find g(-1)

g(-1) = -2 Finally f(-2) is undefined since

division by zero is not allowed.Hence f(g(h(1))) is undefined and x= 1 is not in the domain of f(g(h(x)))

Exercises:

1. Evaluate f(g(3)) given thatf(x) = | x - 6 | + x

2- 1 and g(x) = 2x

2. Find the range of the compositefunction f(g(x)) given thatf(x) = x + 4 and g(x) = x 2+ 2

3. Find the composite function (f og)(x) given thatf = {(3,6) , (5,7) , (9,0)} and g ={(2,3) , (4,5) , (6,7)}

4. Find the composite function (f og)(x) given thatf = {(1,6) , (4,7) , (5,0)} and g ={(6,1) , (7,4) , (0,5)}

Answers to Above Exercises:

1. 35

2. One possibility: f(x) = 2 sec (x) andg(x) = 2x + 1.

3. [0 , 4) U (4 , +infinity)

4. [6 , +infinity)

5. f o g) = {(2 , 6) , (4 , 7)}

6. f o g = {(6 , 6) , (7 , 7) , (0 , 0)}

7)Inverse Function

Solve Using Algebra

work out the inverse using Algebra. Put

"y" for "f(x)" and solve for x:

The function: f(x)= 2x+3

Put "y" for

"f(x)":y= 2x+3

Subtract 3 from

both sides:y-3= 2x

Divide bothsides by 2:

(y-3)/2

= x

Swap sides: x=(y-

3)/2

Solution (put "f-

1(y)" for "x") :f-1(y)=

(y-

3)/2

Here we have the function f(x) = 2x+3,

written as a flow diagram:

The Inverse Functionjust goes the

other way:

So the inverse of: 2x+3 is: (y-3)/2


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Just think ... if there are two or more x-

valuesfor one y-value, how do we know

which one to choose when going back?.

No InverseInverse

is Possible

When a y-value has

more than one x-

value, how do we

know which x-value to

go back to?

When there is a

unique y-value

for every x-value

we can always"go back" from y

to x.

It is called "Injective" or "One-to-

one":

But look at what happens I try to solve for"x=":

My original function:

Solving for "x=":

The inverse is not a function.

Any time you come up with a "" sign, youcan be pretty sure that the inverse isn't afunction.

Find the inverse function ofy= x2+ 1, x< 0.

domain isx< 0and the range (from the graph)

is1 1,and this inverse is also a function.

Here's the graph:


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Example 1 Given find .

Solution

withy.

.

is true.


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Example 2 Given find ,

.

Solution

instead of

, given above is in fact one-to-one.

Without this restriction the inverse would not be one-to-one as is easily seen by a couple of quickevaluations.

Therefore, the restriction is required in order to make sure the inverse is one-to-one.


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There is an interesting relationship between the graph of a function and its inverse.

Here is the graph of the function and inverse from the first two examples.

In both cases , the graph of the inverse is a reflection of the actual function about the

line .

This will always be the case with the graphs of a function and its inverse.

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