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Chapter 2: Problem 77

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). $$ f(x)=\sqrt[3]{2-x} $$

Short Answer

Expert verified
After graphing and analyzing, the function \( f(x)=\sqrt[3]{2-x} \) is one-to-one and has an inverse that is a function.

Step by step solution

01

Graph the function

Using the graphing utility, plot the function \( f(x)=\sqrt[3]{2-x} \). This function is a cube root function, and it is reflected across the y-axis because of the 'minus x'. Further, the graph will pass through the point (2,0).
02

Check if the function is one-to-one

In mathematical terms, a function is said to be one-to-one (or injective), if it assigns each input value to exactly one unique output value. Visually on a graph, a function can be proven to be one-to-one if it passes the horizontal line test. This means that no matter where a horizontal line is drawn, it should always intersect the function's graph at most once.
03

Analyze the resulting graph

From the resulting graph of the function \( f(x)=\sqrt[3]{2-x} \), it can be observed whether it passes the horizontal line test. If it does, the function is one-to-one and hence has an inverse that is also a function.

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