Question

the function h is given by $h(x)=x^{5}+3x - 2$ and $h(1)=2$. if $h^{-1}$ is the inverse of h, what is the value of $(h^{-1})(2)$?
a $\frac{1}{83}$
b $\frac{1}{8}$
c $\frac{1}{2}$
d 1

Explanation:

Step1: Recall the formula for the derivative of an inverse function

If \(y = h(x)\) and \(x=h^{-1}(y)\), then \((h^{-1})'(y)=\frac{1}{h'(x)}\), where \(y = h(x)\).

Step2: Find the derivative of \(h(x)\)

Given \(h(x)=x^{5}+3x - 2\), using the power rule \((x^{n})'=nx^{n - 1}\), we have \(h'(x)=5x^{4}+3\).

Step3: Determine the value of \(x\) when \(y = 2\)

We know that \(h(1)=2\), so when \(y = 2\), \(x = 1\).

Step4: Evaluate \(h'(x)\) at \(x = 1\)

Substitute \(x = 1\) into \(h'(x)\): \(h'(1)=5(1)^{4}+3=5 + 3=8\).

Step5: Use the formula for \((h^{-1})'(y)\)

Since \((h^{-1})'(y)=\frac{1}{h'(x)}\) and \(y = 2,x = 1\), then \((h^{-1})'(2)=\frac{1}{h'(1)}\).
Substitute \(h'(1)=8\) into the formula: \((h^{-1})'(2)=\frac{1}{8}\).

Answer:

B. \(\frac{1}{8}\)