1. Functions and Their Inverses
Essential Questions
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
Holt Algebra2

2. 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.

4. Using the Horizontal-Line Test
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.

5. Using the Horizontal-Line Test
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.

6. Using the Horizontal-Line Test
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.

7. 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.

8. Writing Rules for inverses
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.

9. Writing Rules for inverses
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.

10. Writing Rules for inverses
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.

11. Writing Rules for inverses
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, +µ).

12. Writing Rules for inverses
Find the inverse of f(x) = x2 – 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. 2
+ -
x + 4 = y
Take the square root of both sides.
+ -
x + 4 = y
Simplify.

13. Writing Rules for inverses
Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.
Because the inverse is not a function,
The domain of the inverse is the range of f(x): [ -4, +µ).
The range is the domain of f(x): R.
Check Graph both relations to see that they are symmetric about y = x.

14. Lesson 14.2 Practice A