1. Inverse Functions 5.3 Chapter 5 Functions 5.3.1
MATHPOWERTM 11, WESTERN EDITION
5.3.1

2. The inverse of a function is a reflection of its graph in
Inverse Functions
The inverse of a function is a reflection of its graph in
the line y = x. It can be determined by interchanging
the coordinates of the ordered pairs in the functions.
The inverse of a function is written as f-1(x) and read as
“the inverse of f at x.”
When x and y are interchanged in the equation
of a function:
The coordinates of the points that satisfy the
equation are interchanged.
The graph of the function is reflected in the line y = x.
To determine the inverse of a function:
Interchange x and y in the equation of the function.
Solve the resulting equation for y.
5.3.2

3. Graphing the Inverse Function
Note: If the ordered pair (3, 6) satisfies
the function f(x), then the ordered pair (6, 3)
will satisfy the inverse, f-1(x).
Find the inverse of the function f(x) = 4x - 7.
y = 4x - 7
Interchange the
x and y values.
x = 4y - 7
x + 7 = 4y
x + 7 = y 4
f-1(x)
(-3, 1)
(-7, 0)
y = x
f(x)
(1, -3)
(0, -7)
5.3.3

4. If two functions f(x) and g(x) are inverses of
Verifying an Inverse
If two functions f(x) and g(x) are inverses of
each other, then f(g(x)) must equal x and
g(f(x)) must equal x.
Verify that the functions f(x) = 4x - 7 and
are inverses.
f(g(x)) must be equal to x.
g(f(x)) must be equal to x.
f(x) = 4x - 7
g(4x - 7)
= (x + 7) - 7
= x
= x
Since f(g(x)) and g(f(x)) are both equal to x, then f(x) and g(x)
are inverses of each other.
5.3.4

5. y = x Graphing a Function and Its Inverse Graph f(x) = x2 + 1
(-2, 5)
(2, 5)
The graphs are
symmetrical about
the line y = x.
(1, 2)
For the function:
(-1, 2)
(5, 2)
Domain:
Range:
(2 , 1)
y > 1
(0, 1)
(1, 0)
For the inverse:
Domain:
Range:
x > 1
(2, - 1)
y = x
(5, -2)
Is the inverse
a function?
Could the domain
of f(x) be restricted
so that the inverse
is a function?
5.3.5

6. Graphing a Function and Its Inverse [cont’d]
Given f(x) = x2 + 1,
the inverse is
NOT a function.
Graph y = x2 + 1
where x > 0.
Graph the inverse.
Is this a function?
Graph y = x2 + 1
where x < 0.
Graph its inverse
Is this a function?
What are your
conclusions about
restricting the domain
so that the inverse is
a function?
Vertical Line Test
5.3.6

7. Assignment Questions: Pages p. 99 #3, 5(a,c), 6(a,c,e), 10
and p. 105 #1(a,c), 2(a,c), 4,
5.3.7