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Chapter 1: Problem 67

Determine whether the function has an inverse function. If it does, then find the inverse function. $$h(x)=-\frac{4}{x^{2}}$$

Short Answer

Expert verified
The function \(h(x)=-\frac{4}{x^{2}}\) does not have an inverse function.

Step by step solution

01

Check if the function is one-to-one

A function is one-to-one if no two different elements in the domain of the function map to the same element in the range. The given function \(h(x)=-\frac{4}{x^{2}}\) is not one-to-one. This is evident when considering that both \(h(-2)\) and \(h(2)\) yield \( -4/4 =-1\), thus different values in the domain lead to the same value in the range of the function.
02

Check if the function is onto

A function is onto/surjective if for every element in the codomain, there is a corresponding element in the domain. In this case, there are positive values in the codomain (range) of the function for which no corresponding element in the domain exists, hence the function is not onto.
03

Determine whether the function has an inverse

Since the given function \(h(x)=-\frac{4}{x^{2}}\) is neither one-to-one/injective nor onto/surjective, it does not have an inverse function. Inverse only exist in case of bijective functions, which our function is not.

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