Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=x^{2 / 3}, \quad g(x)=x^{6}\)

Step-by-Step Solution

Answer

The compositions of the functions are (a) \(f \circ g = x^{4}\) and (b) \(g \circ f = x^{4}\)

Step 1: Find \(f \circ g\)

This implies that we must substitute \(g(x)\) into \(f(x)\). So, \(f(g(x)) = f(x^{6}) = (x^{6})^{2 / 3}\), since our \(f\) function raises any input to the power of \(2 / 3\).

Step 2: Simplify \(f \circ g\)

To simplify \( (x^{6})^{2 / 3} \), we have to multiply the exponents according to the rule \( (a^{m})^{n} = a^{mn} \). After doing this, we get: \( (x^{6})^{2 / 3} = x^{(6 * 2 / 3)} = x^{4}\)

Step 3: Find \(g \circ f\)

In a similar way, finding \(g \circ f\) means that we must substitute \(f(x)\) into \(g(x)\). So, \(g(f(x)) = g(x^{2 / 3}) = (x^{2 / 3})^{6}\), since our \(g\) function raises any input to the power of 6.

Step 4: Simplify \(g \circ f\)

To simplify \((x^{2 / 3})^{6}\), we have to multiply the exponents according to the rule \( (a^{m})^{n} = a^{mn} \). After doing this, we get: \((x^{2 / 3})^{6} = x^{(2 / 3 * 6)} = x^{4}\)

Key Concepts

Function Composition

When we talk about function composition, we're referring to the process of combining two functions to create a new function. This is often denoted as \(f \circ g\) or \(g \circ f\), and is read as \