Holt McDougal Algebra 2

Functions and Their InversesFunctions and Their Inverses

Holt Algebra2Holt McDougal Algebra 2

• How do we determine whether the inverse of a function is a function?

• How do we write rules for the inverses of functions?

Holt McDougal Algebra 2

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

1. Use the horizontal-line test to determine whether the inverse of the blue relation is a function.

Using the Horizontal-Line Test

The inverse is a function because no horizontal line passes through two points on the graph.

Holt McDougal Algebra 2

Functions and Their Inverses

2. Use the horizontal-line test to determine whether the inverse of the red relation is a function.

Using the Horizontal-Line Test

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

3. Use the horizontal-line test to determine whether the inverse of the red relation is a function.

Using the Horizontal-Line Test

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 InversesWriting 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 InversesWriting Rules for inverses

4. Find the inverse of . Determine whether it is a function, and state its domain and range.

Rewrite the function using y instead of f(x).

Step 2 Find the inverse.

Simplify.

Switch x and y in the equation.

Cube both sides.

Isolate y.

Holt McDougal Algebra 2

Functions and Their InversesWriting 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 InversesWriting Rules for inverses

5. Find the inverse of f(x) = x2 – 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 InversesWriting Rules for inverses

Rewrite the function using y instead of f(x).

Step 2 Find the inverse.

Take the square root of both sides.

Switch x and y in the equation.

Add 4 to both sides of the equation.

Simplify.

5. Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.

y = x2 – 4

x = y2 – 4

x + 4 = y2

+ - x + 4 = y

2+ - x + 4 = y

Holt McDougal Algebra 2

Functions and Their InversesWriting Rules for inverses

5. Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.

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.

Because the inverse is not a function, . 41 xxf

Holt McDougal Algebra 2

Functions and Their Inverses

Lesson 14.2 Practice A