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

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

Lesson 14.2 Practice A