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…

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 $(h^{-1})(2)$?\na $\frac{1}{83}$\nb $\frac{1}{8}$\nc $\frac{1}{2}$\nd 1

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 $(h^{-1})(2)$?\na $\frac{1}{83}$\nb $\frac{1}{8}$\nc $\frac{1}{2}$\nd 1

Answer

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})