warm-upwarm-up determine and the domain of each section 4.2 one-to-one and inverse functions
TRANSCRIPT
Warm-up
xxg
x
xxf
1
1)(
2)(
Determine and the domain of each gf fg
Section 4.2 One-to-One and Inverse Functions
What did you find for:
What do you think this implies ?
Motivation: Example of inverse functions
These are the functions we use to convert between Celsius & Farenheit
(Celsius to Fahrenheit)
329
5)(
325
9)(
xxg
xxf
xgf
xfg
(Fahrenheit to Celsius)
Section 4.2 One-to-One and Inverse Functions
Inverse Function as ordered pairs
Algebraic Example of an Inverse62)( xxf
Notation : is used to represent the inverse of )(1 xf )(xf
1. Properties of Inverse of a function
Definition: The inverse function of , is called and satisfies the property:
and
IMPORTANT: is NOT
1ff
1
1ff
xxff )(1 xxff )(1
Domain of = Range of
Domain of = Range of
f
f
1f1f
2. Verify inverse functions
State the Domain and Range of f and 1f
52
)( x
xg5
2)(
xxf
Example 1: Prove that f and are inverse functionsg
Example 2 : Verify that f and g are inverses of each other
32
1)( xxg62)( xxf
2. More practice
3. Finding the Inverse of a Function (Switch and solve)
1. Replace f(x) with y
2. Interchange x and y
3. Solve for y
4. Replace y with )(1 xf
Exercises. Find the inverse of each function.
62)( 1) xxf
0 ;1)( 2) 2 xforxxf
45
)( 3) x
xf
3
12)( 4)
x
xxf
4. Domain of the Inverse of a Function
State the domain and range for the function and its inverse.
62)( 1) xxf
0 ;1)( 2) 2 xforxxf
45
)( 3) x
xf
Domain of = Range of
Domain of = Range of
f
f
1f1f
5. Properties of the graph of Inverse
Given the function:
Find the inverse function and complete the table below.
x-interceptsy-interceptsvertical asymptoteshorizontal asymptotes
f 1f
Sketch the graph of both on the same set of axes
3
12)( 4)
x
xxf
6. Symmetry in the graphs
Symmetry: The graph of f -1 and f are symmetric with respect to the line y = x
Points on the graph:If f contains the point (a,b)
then f -1 contains the point (b,a)
6. Practice – Given a graph, sketch its inverse1. Sketch inverse
using symmetry about y = x.
2. Domain of f:
3. Range of f
4. Domain of f -1
5. Range of f –1
7. Determine if a function is One-to-One
Definition: A function is one-to-one if for each y value there is exactly one x value (i.e. y values don’t repeat)
1
1
1
1
Not one-to-one Is one-to-one
7. Horizontal Line Test for Inverse Function
Horizontal Line Test: f is a one-to-one functionif there is no horizontal line that intersects the graph more than once
Definition: f has an inverse that is a function if f passes the horizontal line test
Definition: Domain-Restricted Function: A function’s domain can be restricted to make f one-to-one.
8. Finding the Inverse of a Domain-restricted Function
Example: Restrict the domain of to make it one-to-one.
12)( 2 xxf
Example: Restrict the domain of to make it one-to-one. 1
12)(
x
xxf