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Chapter 1: Problem 58

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\frac{1}{x^{2}}$$

Short Answer

Expert verified
The function \(f(x)=\frac{1}{x^{2}}\) does not have an inverse function.

Step by step solution

01

Determine Whether the Function is Bijective

To find out if the function \(f(x)=\frac{1}{x^{2}}\) is bijective, both injectivity and surjectivity need to be confirmed. For injectivity, it means that for every 'y' in the output there is exactly one 'x' in the input. The function \(f(x)=\frac{1}{x^{2}}\) is not injective since \(f(x)=f(-x)\) always, meaning there are two different inputs, \(x\) and \(-x\), that give the same output, which contradicts the definition of being one-to-one. Therefore, \(f(x)\) is not bijective, and it does not have an inverse function.
02

Finding the Inverse Function (if possible)

Since the function \(f(x)=\frac{1}{x^{2}}\) is not bijective, based on Step 1, it doesn't have an inverse function. Therefore, the process of finding an inverse does not apply.

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