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})

choose the correct graph below. note that the graphs are displayed in a (-5, 10, 1) by (-10, 10, 1) window.

Explanation:

Analyze key points of the function

We are given the function:

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

Let's find key coordinates to identify the correct graph in the window \([-5, 10, 1]\) by \([-10, 10, 1]\):

• \(x\)-intercept: Set \(f(x) = 0 \implies 8 - x = 0 \implies x = 8\). The point is \((8, 0)\).
• \(y\)-intercept: Set \(x = 0 \implies f(0) = \sqrt[3]{8} = 2\). The point is \((0, 2)\).
• Behavior as \(x \to \infty\): \(8 - x \to -\infty \implies f(x) \to -\infty\).
• Behavior as \(x \to -\infty\): \(8 - x \to \infty \implies f(x) \to \infty\).

Match with the options

• The graph must pass through \((0, 2)\) and \((8, 0)\), decreasing from left to right.
• Option A shows a decreasing curve passing through \((0, 2)\) and crossing the \(x\)-axis at \(x = 8\).
• Option B shows an increasing curve.
• Option C shows an increasing curve.
• Option D shows a curve that does not match these intercepts.

Thus, the correct graph is A.

Determine if the function is one-to-one

• A function has an inverse that is a function if and only if it is one-to-one.
• Applying the Horizontal Line Test to the decreasing cube root curve, any horizontal line intersects the graph at most once.
• Therefore, the function is one-to-one and has an inverse that is a function.

Answer:

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