Functions and Their Inverses 6-6 Functions and Their Inverses Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra2

Warm Up Solve for x in terms of y. 1. 2. 3. 4. y = 2ln x

Objectives Determine whether the inverse of a function is a function. Write rules for the inverses of functions.

Vocabulary one-to-one function

Example 1A: 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.

Example 1B: 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.

Check It Out! Example 1 Use the horizontal-line test to determine whether the inverse of each relation is a function. The inverse is a function because no horizontal line passes through two points on the graph.

Example 2: Writing Rules for inverses Find the inverse of . Determine whether it is a function, and state its domain and range.

Example 2 Continued Step 1 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.

Example 2 Continued 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 It Out! Example 2 Find the inverse of f(x) = x3 – 2. Determine whether it is a function, and state its domain and range.

Check It Out! Example 2 Continued Step 1 Find the inverse. y = x3 – 2 Rewrite the function using y instead of f(x). x = y3 – 2 Switch x and y in the equation. x + 2 = y3 Add 2 to both sides of the equation. Take the cube root of both sides. 3 x + 2 = y 3 x + 2 = y Simplify.

Check It Out! Example 2 Continued Because the inverse is a function, . The domain of the inverse is the range of f(x): R. The range is the domain of f(x): R.

You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value. You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compose two inverses the result is the input value x.

Example 3: Determining Whether Functions Are Inverses Determine by composition whether each pair of functions are inverses. 1 3 f(x) = 3x – 1 and g(x) = x + 1 Find the composition f(g(x)). Substitute x + 1 for x in f. 1 3 f(g(x)) = 3( x + 1) – 1 1 3 Use the Distributive Property. = (x + 3) – 1 = x + 2 Simplify.

Example 3 Continued Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)).

Example 3B: Determining Whether Functions Are Inverses For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1 x x – 1 Find the compositions f(g(x)) and g(f (x)). = (x – 1) + 1 = x = x Because f(g(x)) = g(f (x)) = x for all x but 0 and 1, f and g are inverses.

Determine by composition whether each pair of functions are inverses. Check It Out! Example 3a Determine by composition whether each pair of functions are inverses. 3 2 f(x) = x + 6 and g(x) = x – 9 Find the composition f(g(x)) and g(f(x)). f(g(x)) = ( x – 9) + 6 3 2 g(f(x)) = ( x + 6) – 9 2 3 = x – 6 + 6 = x + 9 – 9 = x = x Because f(g(x)) = g(f(x)) = x, they are inverses.

10 x - Check It Out! Example 3b f(x) = x2 + 5 and for x ≥ 0 Find the compositions f(g(x)) and g(f(x)). f(g(x)) = + 5 Substitute for x in f. = x + 25 +5 10 x - Simplify. = x – 10 x + 30

Lesson Quiz: Part I 1. Use the horizontal-line test to determine whether the inverse of each relation is a function. A: yes; B: no

Lesson Quiz: Part II 1. Find the inverse f(x) = x2 – 4. Determine whether it is a function, and state its domain and range. 2. Determine by composition whether f(x) = 3(x – 1)2 and g(x) = +1 are inverses for x ≥ 0.

Lesson Quiz: Part III 3. Determine by composition whether f(x) = 3(x – 1)2 and g(x) = +1 are inverses for x ≥ 0. yes