Question

Show that \(f\) and \(g\) are inverses of each other by verifying that \(f[g(x)]=x\) and \(g[f(x)]=x\). $$ \begin{aligned} &f(x)=4(x+1)^{2 / 3}, \text { where } x \geq-1 \\ &g(x)=\frac{1}{8}\left(x^{3 / 2}-8\right), \text { where } x \geq 0 \end{aligned} $$

Short answer

In summary, upon checking the condition \(f[g(x)] = x\), we found that \(f[g(x)] = 4x \neq x\) for non-negative x values. Thus, the given functions \(f(x) = 4(x+1)^{2/3}\) and \(g(x) = \frac{1}{8}(x^{3/2}-8)\) are not inverses of each other, at least not for the entire specified domain.

Step-by-step solution

Substitute g(x) into f(x)

Replace x in the function f(x) with the entire expression of g(x): \[f[g(x)] = 4[(g(x) + 1)^{\frac{2}{3}}]\] Now substitute the given expression for g(x): \[f[g(x)] = 4\left[\left(\frac{1}{8}\left(x^{\frac{3}{2}} - 8\right) + 1\right)^{\frac{2}{3}}\right]\]

Simplify and check if the result is x

First, simplify within the parentheses: \[\frac{1}{8}\left(x^{\frac{3}{2}}-8\right) + 1 = \frac{1}{8}x^{\frac{3}{2}} - 1 + 1 = \frac{1}{8}x^{\frac{3}{2}}\] So, now the expression is: \[f[g(x)] = 4\left[\left(\frac{1}{8}x^{\frac{3}{2}}\right)^{\frac{2}{3}}\right]\] Next, apply the power rule to exponents: \[4\left(x^{\frac{3}{2}\cdot\frac{2}{3}}\right)= 4\left(x^1\right)\] \[f[g(x)] = 4x\] Given that x is non-negative (x ≥ 0), we have: \[f[g(x)] = 4x \neq x\] Thus, the condition \(f[g(x)] = x\) is not satisfied. Since the first condition is not satisfied, there is no need to check for the second condition \(g[f(x)] = x\). We conclude that f(x) and g(x) are not inverses of each other, at least not for the entire domain of non-negative x values given in the problem statement.