Functions
Functions and
and Their
Their Inverses
Inverses
• How do we determine whether the
inverse of a function is a function?
• How do we write rules for the inverses
of functions?
HoltMcDougal
Algebra2 Algebra 2
Holt
Functions and Their Inverses
In previous lessons, you learned that the inverse of a
function f(x) “undoes” f(x). Its graph is a reflection
across line y = x. The inverse may or not be a
function.
Recall that the vertical-line test can help you
determine whether a relation is a function. Similarly,
the horizontal-line test can help you determine
whether the inverse of a function is a function.
Holt McDougal Algebra 2
Functions and Their Inverses
Holt McDougal Algebra 2
Functions and Their Inverses
Using the Horizontal-Line Test
1. Use the horizontal-line test to determine whether the inverse of
the blue relation is a function.
The inverse is a function
because no horizontal line
passes through two points
on the graph.
Holt McDougal Algebra 2
Functions and Their Inverses
Using the Horizontal-Line Test
2. Use the horizontal-line test to determine whether the inverse of
the red relation is a function.
The inverse is a not a
function because a
horizontal line passes
through more than one
point on the graph.
Holt McDougal Algebra 2
Functions and Their Inverses
Using the Horizontal-Line Test
3. Use the horizontal-line test to determine whether the inverse of
the red relation is a function.
The inverse is a function
because no horizontal line
passes through two points
on the graph.
Holt McDougal Algebra 2
Functions and Their Inverses
Recall from previous lessons that to write the rule for
the inverse of a function, you can exchange x and y
and solve the equation for y. Because the value of x
and y are switched, the domain of the function will
be the range of its inverse and vice versa.
Holt McDougal Algebra 2
Functions and Their Inverses
Writing Rules for inverses
4. Find the inverse of
. Determine
whether it is a function, and state its domain and
range.
Step 1 Graph the function.
The horizontal-line test shows
that the inverse is a function.
Note that the domain and
range of f are all real numbers.
Holt McDougal Algebra 2
Functions and Their Inverses
Writing Rules for inverses
4. Find the inverse of
. Determine
whether it is a function, and state its domain and
range.
Step 2 Find the inverse.
Rewrite the function using y instead of f(x).
Switch x and y in the equation.
Cube both sides.
Simplify.
Isolate y.
Holt McDougal Algebra 2
Functions and Their Inverses
Writing Rules for inverses
4. Find the inverse of
. Determine
whether it is a function, and state its domain and
range.
Because the inverse is a function,
.
The domain of the inverse is the range of
f(x):{x|x  R}.
The range is the domain of f(x):{y|y R}.
Check Graph both relations to see that they are
symmetric about y = x.
Holt McDougal Algebra 2
Functions and Their Inverses
Writing Rules for inverses
2
5. Find the inverse of f(x) = x – 4. Determine
whether it is a function, and state its domain and
range.
Step 1 Graph the function.
The horizontal-line test shows
that the inverse is not a
function. Note that the domain
of f is all real numbers but the
range is [ -4, +.
Holt McDougal Algebra 2
Functions and Their Inverses
Writing Rules for inverses
2
5. Find the inverse of f(x) = x – 4. Determine
whether it is a function, and state its domain and
range.
Step 2 Find the inverse.
y = x2 – 4
Rewrite the function using y instead of f(x).
x = y2 – 4
Switch x and y in the equation.
x + 4 = y2
Add 4 to both sides of the equation.
+
-
x + 4 = y2
Take the square root of both sides.
+
-
x+4=y
Simplify.
Holt McDougal Algebra 2
Functions and Their Inverses
Writing Rules for inverses
2
5. Find the inverse of f(x) = x – 4. Determine
whether it is a function, and state its domain and
range.
-1
Because the inverse is not a function, f x    x + 4.
The domain of the inverse is the range of f(x): [ -4, +.
The range is the domain of f(x): R.
Check Graph both relations to see that they
are symmetric about y = x.
Holt McDougal Algebra 2
Functions and Their Inverses
Lesson 14.2 Practice A
Holt McDougal Algebra 2