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Chapter 1: Problem 31

Find (a) \(f \circ g,\) (b) \(g \circ f,\) and (c) \(g \circ g\). $$f(x)=x^{2}, \quad g(x)=x-1$$

Short Answer

Expert verified
The composite functions (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(g \circ g\) are \(f(g(x))=(x-1)^{2}\), \(g(f(x))=(x^{2}) - 1\), and \(g(g(x))=x-2\), respectively.

Step by step solution

01

Finding \(f \circ g\)

The composite function \(f \circ g\) means 'f of g or f composed with g'. It is evaluated as \(f(g(x))\). Substituting \(g(x)\) into \(f(x)\) by replacing every \(x\) in \(f(x)\) with \(g(x)\) gives \(f(g(x)) = (x-1)^{2}\). So, \(f(g(x))=(x-1)^{2}\) is the function \(f \circ g\).
02

Finding \(g \circ f\)

The composite function \(g \circ f\) means 'g of f or g composed with f'. It is evaluated as \(g(f(x))\). Substituting \(f(x)\) into \(g(x)\) by replacing every \(x\) in \(g(x)\) with \(f(x)\) gives \(g(f(x)) = (x^{2}) - 1\). So, \(g(f(x))=(x^{2}) - 1\) is the function \(g \circ f\).
03

Finding \(g \circ g\)

The composite function \(g \circ g\) means 'g of g or g composed with g'. It is evaluated as \(g(g(x))\). Substituting \(g(x)\) into \(g(x)\) again by replacing every \(x\) in \(g(x)\) with \(g(x)\) gives \(g(g(x)) = (x-1) - 1 = x-2\). So, \(g(g(x))= (x-2)\) is the function \(g \circ g\).

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Most popular questions from this chapter

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