MHF4U
Inverse Functions
The _________________ of a function is the result of exchanging the input(x) and output(y) values of
the function. An inverse is the _______________ of the original function or relation.
Graphically
If you are given the graph of a function,
1. reverse the coordinates of each point (imagine how a table of values would compare)
2. plot these points
3. sketch the relation
y x
y=x2
eg.
Y=
































Notice that the graph of the inverse is a reflection of the
original graph in the
line. The graphs are
symmetical.




y=x2



An inverse function has the same properties as an original
function (for each x there is only one value of y).










y x


The domain of the original = range of the inverse, and
the range of the original = the domain of the inverse.



NOTE: Not all functions have an inverse that is a function, although it is possible to restrict the
domain of the original function so that the inverse is a function.
e.g.
We can only use the notation f--1(x) if the inverse is a function. (Note: f--1(x) ≠
1
f
 not an exponent!)
Algebraically
If a function is represented by an equation in x and y, then the inverse of the function is found by
exchanging the input(x) and output(y) values. (Simply, switch ___ and ___)
Example 1 Find f-1(x), the inverse of f(x) = 3x – 2.
1. Write the function in terms of x and y.
y = 3x - 2
2. Exchange the x and the y.
x = 3y - 2
3. Solve for y, so that the relation can be written in function notation.
3y – 2 = x
3y  x  2
x2
y
3
x2
f 1 ( x) 
3
Example 2 Find the inverse of f(x)  x2  4x .
y  x2  4x
The inverse is:
x  y2  4y
To solve for y, write the relation as a quadratic equation.
y2  4y  x  0
Solve using the quadratic formula.
y
y
y
y
b  b 2  4ac
2a
4  42  4 1  x 
2 1
4  16  4 x
2
4  4  4  x 
2
4  2 4  x
y
2
y  2  4  x
**Notice that the inverse is not a function, so it cannot be written in function notation.
1.7 – Inverses of Functions: Homework
In questions 1 and 2, write a table that represents the inverse of the function given by the table and state whether or
not the inverse is a function.
1.
x
1
2
3
4
5
2.
f(x)
4
2
3
6
1
x
-1
0
1
2
3
f(x)
4
3
4
1
5
In questions 3 and 4, the graph of a function f is given. Sketch the graph of the inverse function of f and give the
coordinates of three points on the inverse.
3
4.
In questions 5-18, find the rule for the inverse of the given function. Solve your answers for y and, if possible, write
in function notation.
5. f(x)  x
6. f(x)  x  1
3
9. f(x)  5  2x
5
10. f(x)  x  1
13. f(x) 
1
x
17. f(x) 
x 1
x3  5

18. f(x) 
3
1
x
14. f(x) 
3

5
2
7. f(x)  5x  4
2
8. f(x)  3x  5
11. f(x)  4x  7
12. f(x)  5  3x  2
15. f(x) 
1
2x  1
2
16. f(x) 
1
x 1
2
3x  1
x2
In questions 19-23, each given function has an inverse function. Sketch the graph of the inverse function.
19. f(x)  x  3
20. f(x)  3x  2
 x2  1
for x  0
0.5x  1 for x  0
23. f(x)  
5
21. f(x)  0.3x  2
22. f(x)  3 x  3
In questions 24-31, none of the functions has an inverse that is a function. State at least one way of restricting the
domain of the function so that the restricted function has an inverse that is a function. Then find the rule of the
inverse function.
24. f(x)  x
28. f(x) 
x2  6
2
25. f(x)  x  3
2
26. f(x)   x
29. f(x)  4  x2
30. f(x) 
32. Show that the inverse function f whose rule is f(x) 
1
x 1
2
2
27. f(x)  x  4
31. f(x)  3  x  5   2
2
2x  1
is f itself.
3x  2
33. List three different functions (other than the one in 32.), each of which is its own inverse. There are many correct
answers.
34. Let m and b be constants, with m  0 , Find the inverse function, f -1 of f(x)=mx + b.
35. Show that the points P=(a, b) and Q=(b, a) are symmetric with respect to the line y = x as follows:
a) Find the slope of the line through P and Q.
b) Use slopes to show that the line through P and Q is perpendicular to y = x.
c) Let R be the point where the line y=x intersects PQ. Since R is on y=x, it has coordinates (c, c) for some number c.
Use the distance formula to show that PR has the same length as RQ. Conclude that the line y=x is the
perpendicular bisector of PQ. Therefore, P and Q are symmetric with respect to the line y=x.
y=x

P




Q









Answers
1.
y
4
2
3
6
1
Function
2.
f(y)
1
2
3
4
5
y
f(y)
4
-1
3
0
4
1
1
2
5
3
Not a function
3.
4.
5.
f 1 ( x)   x
8.
y
6. f 1 ( x)  1  x
5 x
3
9. f 1 ( x) 
x2  7
,
11. f ( x) 
4
1
14. f 1 ( x)  2
x
1
17. f 1 ( x) 
3
 x  0
34. f 1 ( x) 
5 x
2
 x  5
( x) 
2
1 x
2x
2 x5  1
1
18. f ( x)  5
x 3
5x  1
1 x
2
3
15. y  
19.
24. x  0
29. x  0
12. f
1
3
7. y  
xb
m
25. x  3
30. x  0
35. a) –1
10. f 1 ( x) 
5 3
13. f 1 ( x) 
1
x
16. y  
21.
x4
5
x 1
1 x
x
23.
26. x  0
31. x  5
27. x  0
28. x  0
b) slopes are negative reciprocals or product is –1