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Chapter 2: Problem 96

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{6}-5 $$

Short Answer

Expert verified
The inverse function of \(f(x) = x^6 - 5\) is \(f^{-1}(x) = \sqrt[6]{x + 5}\).

Step by step solution

01

Rewrite the function with the inverse notation

Replace \(f(x)\) with \(y\). This will help us to isolate x and later swap x and y to get the inverse function notation. \(y = x^6 - 5\)
02

Swap x and y variables

We will now switch the positions of x and y in the equation both sides: \(x = y^6 - 5\)
03

Solve for y

Rearrange the equation to solve for y: \(x+5 = y^6\) Now, find the sixth root on both sides of the equation to isolate y: \(f^{-1}(x) = y = \sqrt[6]{x + 5}\)
04

Write the inverse function

Now we can write the inverse function \(f^{-1}(x)\) as: \(f^{-1}(x) = \sqrt[6]{x + 5}\) The inverse function of \(f(x) = x^6 - 5\) is \(f^{-1}(x) = \sqrt[6]{x + 5}\).

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Most popular questions from this chapter

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{9 x+5}{x^{2}-x-6} $$

Write the indicated expression as \(a\) polynomial. $$ (p(x))^{2} s(x) $$

Suppose \(q\) is a polynomial of degree 5 such that \(q(1)=-3\). Define \(p\) by $$ p(x)=x^{6}+q(x) $$ Explain why \(p\) has at least two zeros.

Suppose \(q\) is a polynomial of degree 4 such that $$ \begin{array}{r} q(0)=-1 . \text { Define } p \text { by } \\ \qquad p(x)=x^{5}+q(x) . \end{array} $$ Explain why \(p\) has a zero on the interval \((0, \infty)\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} t(x) $$
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