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Chapter 1: Problem 103

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, the restricted function \(f(x) = x^{2} + 1\), for all \(x \geq 0\), has an inverse because it is a one-one function.

Step by step solution

01

Restrict the Domain

We are given function \(f(x) = x^{2} + 1\). We'll restrict its domain to \(x \geq 0\). Now the function is \(f(x) = x^{2} + 1\), for all \(x \geq 0\).
02

Graphing the Function

Now we'll graph the function with the restricted domain. As \(x^{2}\) will always be positive and 1 is added to it, the graph starts from 1 (y-axis) and moves through the first and the second quarters, as x is \( \geq 0.\) The graph is the right half of a U shape (parabola opening upwards) below the x-axis.
03

Find out if there is an Inverse

For a given function, if every 'x' in the domain maps to a unique 'y' in the range, and for every 'y' in the range there is only one 'x' in the domain, then that function is said to have an inverse. Here in our graph for every 'x' in the domain there is only one 'y' in the range meaning it is a one-one function. For every 'y' in the range there is only one 'x' in the domain. So, we can say that the function has an inverse.

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