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Chapter 2: Problem 37

Find a formula for \(f\) o \(g\) given the indicated functions \(f\) and \(g\). $$ f(x)=4 x^{-2}, g(x)=5 x^{3} $$

Short Answer

Expert verified
The formula for the composition of functions \(f(g(x))\) or \((f \circ g)(x)\) is given by: \((f \circ g)(x) = \frac{4}{25}x^{-6}\).

Step by step solution

01

Understand the composition of functions

A composition of functions, \((f \circ g)(x)\), can be expressed as \(f(g(x))\). In order to find the formula for the composition, we need to replace every occurrence of \(x\) in the function \(f(x)\) with the expression for the function \(g(x)\).
02

Substitute \(g(x)\) in the expression of \(f(x)\)

In this step, we will replace \(x\) in the expression for \(f\) with the expression for \(g\). The function \(f\) is given as \(f(x) = 4x^{-2}\), and the function \(g\) is given as \(g(x) = 5x^3\). So, in the expression for \(f\), we will replace \(x\) with \((5x^3)\): $$ f(g(x)) = 4(5x^3)^{-2} $$
03

Simplify the expression

Now, we need to simplify the expression in the exponent. Using the rule of exponents, \((a^m)^n = a^{mn}\), we get: $$ f(g(x)) = 4(5^{-2}x^{-6}) = 4 \cdot 5^{-2}x^{-6} $$
04

Complete the simplification

Finally, we simplify the expression involving the constant terms: $$ f(g(x)) = 4 \cdot \frac{1}{25}x^{-6} = \frac{4}{25}x^{-6} $$ Thus, the composition of functions \(f(g(x))\) is given by: $$ (f \circ g)(x) = \frac{4}{25}x^{-6} $$

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