6-7 Inverse Relations and Functions
Quick Review
If a relation or a function is described by an equation in x and y, you can interchange x and y to get the inverse. The domain of a function becomes the range of its inverse, and the range of a function becomes the domain of its inverse.
Example
What is the inverse of
f
(
x
)
=
x
−
10
?
f , open x close , equals , square root of x minus 10 end root , question mark
y
=
x
−
10
Rewrite using
y
.
x
=
y
−
10
Interchange the
x
and
y
values
.
x
2
=
y
−
10
Square each side
.
y
=
x
2
+
10
Solve for
y
.
f
−
1
(
x
)
=
x
2
+
10
Write the inverse function
.
table with 5 rows and 2 columns , row1 column 1 , y equals , square root of x minus 10 end root , column 2 cap rewriteusing . y . , row2 column 1 , x equals , square root of y minus 10 end root , column 2 cap interchangethe . x , and , y , values , . , row3 column 1 , x squared , equals y minus 10 , column 2 cap squareeachside . . , row4 column 1 , y equals , x squared , plus 10 , column 2 cap solvefor . y . , row5 column 1 , f super negative 1 end super . open x close , equals , x squared , plus 10 , column 2 cap writetheinversefunction . . , end table
The domain of f(x) is
x
≥
10
,
x greater than or equal to 10 comma which means the range of
f
−
1
(
x
)
f super negative 1 end super , open x close is
y
≥
10
.
y greater than or equal to 10 . Also, since the range of f(x) is
y
≥
0
,
y greater than or equal to 0 comma the domain of
f
−
1
(
x
)
f super negative 1 end super , open x close is
x
≥
0
.
x greater than or equal to 0 .
Exercises
Find the inverse of each function. Determine whether each inverse is a function.
-
f
(
x
)
=
2
x
2
−
8
f open x close equals , 2 x squared , minus 8
-
f
(
x
)
=
15
−
3
x
f open x close equals 15 minus 3 x
-
f
(
x
)
=
x
+
6
f , open x close , equals , square root of x plus 6 end root
-
f
(
x
)
=
(
2
x
−
3
)
2
f open x close equals . open 2 x minus 3 close squared
Graph each function and its inverse. Describe the domain and range of each.
-
f
(
x
)
=
4
x
−
1
f open x close equals 4 x minus 1
-
f
(
x
)
=
(
x
+
3
)
2
f open x close equals . open x plus 3 close squared
-
f
(
x
)
=
x
−
3
f , open x close , equals , square root of x minus 3 end root
-
f
(
x
)
=
6
−
5
x
2
f open x close equals 6 minus , 5 x squared
-
Geometry The volume of cube is determined by the formula
V
=
s
3
,
v equals , s cubed , comma where s is the length of one side. Find the inverse formula. Use it to find the side length of a cube with a volume of
64
ft
3
.
64 , ft cubed . .
6-8 Graphing Radical Functions
Quick Review
The function
f
(
x
)
=
x
f , open x close , equals square root of x is the parent function of the square root function
f
(
x
)
=
a
x
−
h
+
k
.
f , open x close , equals eh , square root of x minus h end root , plus k . The graph of
f
(
x
)
=
a
x
f , open x close , equals eh square root of x is a stretch (
a
>
1
eh greater than 1 ) or a shrink
(
0
<
a
<
1
)
open 0 less than eh less than 1 close of the parent function. The graph of
f
(
x
)
=
a
x
−
h
+
k
f , open x close , equals eh , square root of x minus h end root , plus k is a translation h units horizontally and k units vertically of
y
=
a
x
.
y equals eh square root of x . The graph of
f
(
x
)
=
x
n
f , open x close , equals , the th , root of x , is transformed by a, h, and k in the same way as the graph of
f
(
x
)
=
x
.
f , open x close , equals square root of x .
Example
Describe the graph of
y
=
4
x
+
12
.
y equals . square root of 4 x plus 12 end root . .
y
=
4
x
+
12
y
=
4
(
x
+
3
)
Factor the polynomial
.
y
=
2
x
+
3
Simplify the radical
.
table with 3 rows and 2 columns , row1 column 1 , y equals . square root of 4 x plus 12 end root , column 2 , row2 column 1 , y equals . square root of 4 . open , x plus 3 , close end root , column 2 cap factorthepolynomial . . , row3 column 1 , y equals 2 , square root of x plus 3 end root , column 2 cap simplifytheradical . . , end table
The graph of
y
=
4
x
+
12
y equals . square root of 4 x plus 12 end root is the graph of
y
=
2
x
y equals 2 square root of x translated 3 units to the left.
Exercises
Graph each function. Find the domain and range.
-
y
=
x
−
5
y equals square root of x minus 5
-
y
=
x
+
8
y equals , square root of x plus 8 end root
-
y
=
5
x
+
9
y equals 5 square root of x plus 9
-
y
=
−
x
−
4
y equals negative , square root of x minus 4 end root
-
y
=
x
+
10
3
y equals . cube root of x plus 10 end root ,
-
y
=
−
x
−
2
3
+
5
y equals negative . cube root of x minus 2 end root , . plus 5
Rewrite each function to make it easy to graph using transformations. Describe each graph.
-
y
=
9
x
−
27
+
4
y equals . square root of 9 x minus 27 end root . plus 4
-
y
=
−
3
4
x
−
16
y equals negative 3 . square root of 4 x minus 16 end root
-
y
=
8
x
+
24
3
y equals . cube root of 8 x plus 24 end root ,
-
y
=
x
−
4
4
+
6
y equals . square root of fraction x minus 4 , over 4 end fraction end root . plus 6
Solve each equation by graphing.
-
5
=
−
x
−
3
5 equals negative , square root of x minus 3 end root
-
8
x
−
16
=
2
x
+
2
square root of 8 x minus 16 end root . equals 2 , square root of x plus 2 end root
