xgf  xfg  f . If xgf  = x and xfg  = x, then

Name______________________________________
Inverse Functions
Warm-Up: Given f(x) = 3x + 2 and g ( x) 
a) f  g x 
Date____________
10H Per_________
x2
, find:
3
b) g  f x 
If f  g x  = x and g  f x  = x, then _______________________________
______________________________________________________________
The _______________ of a function is the set of ordered pairs obtained by interchanging the _____ and _______ elements
of each pair in the original function.
 For the function f(x), the inverse function is denoted by f 1 ( x) (pronounced “f inverse of x”)
Ex. 1: a) Given the function f = {(0, 1), (1, 2), (2, 3)}, find f
b) Is f
1
1
.
a function? Why or why not?
c) What does this tell us about the original function?
If a function is _____________________________, then its inverse will also be a function.
You can find an inverse by simply ________________________________, as in examples two and three:
Ex. 2: If f
1
= {(-2, 3), (9, 1), (0, -3), (-7, 6)}, find f.
Ex. 3: Determine the inverse of the function. Is its inverse a function?
x
f (x)
x
f -1(x)
1
2
-2
0
-1
3
0
-1
2
1
3
-2
4
5
-3
1
You can also find a functions inverse by __________________________________.
Ex. 4: Find the inverse of the function f(x) = 4x +9
Finding an inverse algebraically
1.
2.
3.
Ex. 5: Find the equation of y  3 x  4 after it is reflected over y = x.
Ex. 6: If g ( x) 
x7
, find g-1(3).
2
If a function is composed with its inverse, the result is _________________________.
o f 1 ( f (2))  2
o
( g  g 1 )( x)  x
The graph of an inverse relation is the reflection of the original graph over the identity line, _________.
*Not all graphs produce an inverse relation which is also a function.*
To sketch an inverse of a given graph, simple swap the ____________________________ and plot the new points.
Ex. 7: The accompanying graph shows the relationship between the cooling
time of magma and the size of the crystals produced after a volcanic eruption.
On the same graph, sketch the inverse of this function.
Ex. 8: Consider the graphs below. The original function is drawn as a solid line. It is then reflected over the identity line, y = x,
and the new dotted graph is the inverse relation.
a) Determine if each inverse is a function.
b) Provide a reason why or why not.
c) Find each inverse algebraically
i)
ii)
a) ___________________________
a) ___________________________
b) ___________________________
b) ___________________________
c) ___________________________
c) ___________________________
Homework:
1. A function is defined by the equation y = 5x – 5. What is the equation that defines the inverse of this function?
2. What is an equation of the line formed when the line y = 3x + 1 is reflected in the line y = x?
3. Given set A: {(1, 2), (2, 3), (3, 4), (4, 5)}, if the inverse is A-1, which statement must be true?
a) A and A-1 are functions.
b) A nor A-1 are functions.
-1
c) A is a function and A is not a function.
d) A is not a function and A-1 is a function.
4. Given the relation A: {(3, 2), (5, 3), (6, 2), (7, 4)}, which statement is true?
a) A and A-1 are functions.
b) A nor A-1 are functions.
c) A is a function and A-1 is not a function.
d) A is not a function and A-1 is a function.
5. Which graph has an inverse that is a function?
a)
b)
c)
d)
6. Find the inverse of the following functions.
a) f(x) = 3x + 8
b) f(x) = x2 + 10
c) g(x) = (x + 4)3
7. State the domain (with restrictions) of each function:
b) h( x) 
a) f ( x)   2 x  8
4
x  7 x  10
2
c) g ( x) 
x
3x  9
8. If f(x) = 2x + 6 and g ( x)  x 2  2 , find the values of:
a) ( f  g )( x)
b) g ( f (2))
c) ( f  f
1
)(3)
9. The function, f, is drawn on the accompanying set of axes. On the same set of axes, sketch the graph of f
inverse of f.
10. Use composition to prove that f ( x) 
than this!).
1
, the
3x  7
2x  7
and g ( x) 
are inverses of each other (you will need more room
2
3

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