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Inverse Relations and
Functions
OBJ:  Find the inverse of a relation
 Draw the graph of a function
and its inverse
 Determine whether the
inverse of a function
is a function
FINDING INVERSES OF LINEAR FUNCTIONS
An inverse relation maps the output values back to their
original input values. This means that the domain of the
inverse relation is the range of the original relation and
that the range of the inverse relation is the domain of the
original relation.
Original relation
x
y
– 2 DOMAIN
–1 0 1
4
Inverse relation
2
x
RANGE
2 0 –2 –4
y
4 DOMAIN
2 0 –2 –4
– 2 RANGE
–1 0
1
2
FINDING INVERSES OF LINEAR FUNCTIONS
Original relation
x –2 –1 0
y
4
2
1
Inverse relation
2
0 –2 –4
Graph of original
relation
Reflection in y = x
Graph of inverse
relation
x
4
2
0 –2 –4
y –2 –1 0
1
2
y=x
FINDING INVERSES OF LINEAR FUNCTIONS
To find the inverse of a relation that is given by an
equation in x and y, switch the roles of x and y and
solve for y (if possible).
Finding an Inverse Relation
Find an equation for the inverse of the relation y
= 2 x – 4.
SOLUTION
y=2x–4
x =2y – 4
x + 4 = 2y
1x+2=y
2
Write original relation.
Switch x
x and yy.
Add 4
4 to each side.
Divide each side by 2.
2
The inverse relation is y = 1 x + 2.
2
If both the original relation and the inverse relation happen to be
functions, the two functions are called inverse functions.
Finding an Inverse Relation
INVERSE FUNCTIONS
Functions
f
and
g are inverses of each other provided:
f (g (x)) = x
g ( f (x)) = x
and
The function g is denoted by
f
–1
, read as “f inverse.”
Given any function, you can always find its inverse relation
by switching x and y.
For a linear function f (x ) = mx + b where m  0, the
inverse is itself a linear function.
Verifying Inverse Functions
Verify that f (x) = 2x – 4 and g (x) =
1 x + 2 are inverses.
2
SOLUTION
Show that f (g (x)) = x and g (f (x)) = x.
1
f (g (x)) = f 2 x + 2
1
= 2 x +2 –4
2
= x+4 – 4
(
(
=x
)
)
g (f (x)) = g (2x – 4)
= 1 (2x – 4) + 2
2
= x–2+2
=x
FINDING INVERSES OF NONLINEAR FUNCTIONS
Finding an Inverse Power Function
Find the inverse of the function f (x) = x 2.
SOLUTION
f (x) = x 2
Write original function.
y = x2
Replace original
x = y2
Switch x and y.
± x=y
f (x) with y.
Take square roots of each side.
x0
FINDING INVERSES OF NONLINEAR FUNCTIONS
The graphs of the power functions f (x) = x 2 and g (x) = x 3 are
shown along with their reflections in the line y = x.
Notice that the inverse of g (x) = x 3 is a function, but that
the inverse of f (x) = x 2 is not a function.
On the other hand, the graph of
g (x) = xf3(xcannot
) = x 2 be intersected
twice with a horizontal line and
its inverse is a function.
gof
(
xf)(x
=
x=3 x 2
Notice
that
the
graph
)
3
–1
g
(
x
)
=
x
can be intersected
twice with a
horizontal line and that its inverse
is not a function.
x= y2
If the domain of f (x) = x 2 is restricted, say to only nonnegative
numbers, then the inverse of f is a function.
FINDING INVERSES OF NONLINEAR FUNCTIONS
H O R I Z O N T A L L I N E T E ST
If no horizontal line intersects the graph
of a function f more than once, then the
inverse of f is itself a function.
Modeling with an Inverse Function
ASTRONOMY Near the end of a
star’s life the star will eject gas,
forming a planetary nebula. The
Ring Nebula is an example of a
planetary nebula.
The volume V (in cubic kilometers) of this nebula
can be modeled by V = (9.01 X 10 26 ) t 3 where t
is the age (in years) of the nebula. Write the
inverse function that gives the age of the nebula
as a function of its volume.
Modeling with an Inverse Function
Volume V can be modeled by V = (9.01 X 10 26 ) t 3
Write the inverse function that gives the age of the nebula
as a function of its volume.
SOLUTION
V = (9.01 X 1026 ) t 3
V
9.01 X 10
3
26
V
9.01 X 10 26
= t3
=t
(1.04 X 10– 9 ) V = t
3
Write original function.
Isolate power.
Take cube root
of each side.
Simplify.
Modeling with an Inverse Function
Determine the approximate age of the Ring Nebula given
that its volume is about 1.5 X 10 38 cubic kilometers.
SOLUTION
To find the age of the nebula, substitute 1.5 X 10 38 for V.
t = (1.04 X 10–9 ) 3 V
Write inverse function.
= (1.04 X 10–9 ) 3 1.5 X 1038
Substitute for V.
 5500
Use calculator.
The Ring Nebula is about 5500 years old.