Question

5.1.61
2 parts -- of 1 point

use the graph of f(x) to determine whether the function is one-to-one. if it is, find a formula for its inverse.

(f(x) = \frac{x + 6}{x - 7})

part 1 of 2
1 try left

Explanation:

Analyze the function type

Using the Rational Functions knowledge point

$$ f(x) = \frac{x + 6}{x - 7},\quad x e 7 $$

Determine if the function is one-to-one

Using the One-to-One Functions and Horizontal Line Test knowledge points

$$
LATEXBLOCK0
$$

The function is one-to-one.

Set up the inverse equation

Using the Inverse Functions knowledge point

$$ y = \frac{x + 6}{x - 7} \implies x = \frac{y + 6}{y - 7} $$

Solve for the inverse function

Using the Inverse Functions knowledge point

$$
LATEXBLOCK1
$$

Answer:

The function is one-to-one. The formula for its inverse is:

$$f^{-1}(x) = \frac{7x + 6}{x - 1}$$