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Chapter 1: Problem 148

Let \(f(x)=1+x^{2}\). Find a function \(g(x)\) such that \(f(g(x))=1+x^{2}-2 x^{3}+x^{4}\)

Short Answer

Expert verified
The function \(g(x)\) is given by \(\sqrt{x^{2}-2x^{3}+x^{4}}\).

Step by step solution

01

Understand the function composition

Firstly, recall that the function composition \(f(g(x))\) means that wherever \(x\) appears in \(f\), it'll be replaced by \(g(x)\). In this case, given that \(f(x)=1+x^{2}\) it means that \(f(g(x))=1+g(x)^{2}\).
02

Compare the compositions

We are given \(f(g(x))=1+x^{2}-2x^{3}+x^{4}\). Comparing this with our equation from Step 1, i.e., \(f(g(x))=1+g(x)^{2}\), implies that the part \(x^{2}-2x^{3}+x^{4}\) is corresponding to \(g(x)^2\).
03

Solve for \(g(x)\)

From step 2 we get \(g(x)^2=x^{2}-2x^{3}+x^{4}\). Hence, \(g(x)\) can be concluded as being \(\sqrt{x^{2}-2x^{3}+x^{4}}\) or \(-\sqrt{x^{2}-2x^{3}+x^{4}}\). However, usually, the main branch of the root function is considered, which is the positive one.

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Most popular questions from this chapter

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