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Chapter 1: Problem 115

Restrict the domain of \(f(x)=x^{2}+1\) to \(x \geq 0 .\) Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.

Short Answer

Expert verified
Yes, by restricting the domain of the function \(f(x)=x^{2}+1\) to \(x \geq 0\), the function has an inverse. This is because the function is one-to-one and passes the horizontal line test within the restricted domain.

Step by step solution

01

Restrict the function

Restrict the domain of the function \(f(x) = x^2 + 1\) to \(x \geq 0\). So, the function will look like \(f(x)=x^{2}+1, x \geq 0\)
02

Graph the function

Plot function \(f(x)=x^{2}+1\) for \(x \geq 0\). The graph of the function will show that it's monotonically increasing and continuous on the interval \(x \geq 0\). The graph starts at the point (0,1) and extends upwards and towards the right.
03

Determine the Inverse

Since the graph of \(f(x)=x^{2}+1, x \geq 0\) passes the horizontal line test, it has an inverse.
04

Explain the Reason

The inverse of a function exists only if the function is one-to-one. One-to-one functions pass the horizontal line test, meaning at no point does the function 'double back' on itself. We restricted the domain to \(x \geq 0\) to guarantee the function will be one-to-one and have an inverse.

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