Question

use a graphing utility to graph the function. use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one).

(f(x) = \sqrt3{8 - x})

use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one.) select the correct choice below.

• the function is one-to-one. it has an inverse.
• the function is not one-to-one. there is no inverse function.

Explanation:

Analyze the function behavior

Using the Cube Root Functions and Graphing Radical Functions knowledge points

$$ f(x) = \sqrt[3]{8-x} $$

• Domain: \((-\infty, \infty)\)
• \(y\)-intercept: \(f(0) = \sqrt[3]{8} = 2\)
• \(x\)-intercept: \(f(x) = 0 \implies 8-x = 0 \implies x = 8\)
• As \(x \to \infty\), \(f(x) \to -\infty\)
• As \(x \to -\infty\), \(f(x) \to \infty\)
• The function is strictly decreasing.

Identify the correct graph

Using the Graphing Radical Functions knowledge point

• Graph A shows a decreasing curve passing through \((0, 2)\) and \((8, 0)\).
• Graph B shows an increasing curve.
• Graph C shows an increasing curve.
• Graph D shows a decreasing curve, but with incorrect scaling/intercepts compared to the standard transformation.
• Therefore, the correct graph is A.

Apply the horizontal line test

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

• Since \(f(x) = \sqrt[3]{8-x}\) is strictly decreasing, any horizontal line intersects the graph at most once.
• Thus, the function is one-to-one and has an inverse function.

Answer:

Question 1

• (A) Decreasing curve passing through (0, 2) and (8, 0) (Correct answer)
• (B) Increasing curve
• (C) Increasing curve
• (D) Decreasing curve with incorrect intercepts

Question 2

• The function is one-to-one. It has an inverse. (Correct answer)
• The function is not one-to-one. There is no inverse function.