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Chapter 2: Problem 95

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=6+x^{3} $$

Short Answer

Expert verified
The inverse function of the given function \(f(x) = 6 + x^3\) is \(f^{-1}(x) = \sqrt[3]{x - 6}\).

Step by step solution

01

Replace function notation with y

Replace \(f(x)\) with \(y\) to make the equation easier to manipulate: $$ y = 6 + x^3 $$
02

Swap the roles of x and y

Swap the x and y variables in the equation: $$ x = 6 + y^3 $$
03

Solve the equation for y

We want to get y by itself on one side of the equation, so first subtract 6 from both sides: $$ x - 6 = y^3 $$ Now take the cube root of both sides to isolate y: $$ y = \sqrt[3]{x - 6} $$
04

Write the resulting equation using the inverse function notation

Replace y with the inverse function notation \(f^{-1}(x)\): $$ f^{-1}(x) = \sqrt[3]{x - 6} $$ The inverse function is given by: $$ f^{-1}(x) = \sqrt[3]{x - 6} $$

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Most popular questions from this chapter

Suppose \(t(x)=\frac{5}{4 x^{3}+3}\). (a) Show that the point (-1,-5) is on the graph of \(t\) (b) Give an estimate for the slope of a line containing (-1,-5) and a point on the graph of \(t\) very close to (-1,-5)

Suppose $$r(x)=\frac{x+1}{x^{2}+3} \quad \text { and } \quad s(x)=\frac{x+2}{x^{2}+5}$$ What is the domain of \(r ?\)

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{6}+3 x^{3}+1}{x^{2}+2 x+5} $$

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Explain why the polynomial \(p\) defined by $$ p(x)=x^{6}+100 x^{2}+5 $$ has no real zeros.
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