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Chapter 2: Problem 91

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{-2 / 5} $$

Short Answer

Expert verified
The inverse function of \(f(x) = x^{-\frac{2}{5}}\) is \(f^{-1}(x) = x^{-\frac{5}{2}}\).

Step by step solution

01

Write the function in terms of y

Write the given function \(f(x) = x^{-2/5}\) as \(y = x^{-2/5}\).
02

Swap x and y

Swap \(x\) and \(y\) so that we can find the inverse function in terms of \(x\). After swapping, the equation will be \(x = y^{-2/5}\).
03

Solve for y

To solve for \(y\), first raise both sides to the power of \(-5/2\) to remove the exponent on the right-hand side: \[ (x^{-1})^{\frac{-5}{2}}=y^{\frac{-5}{2}\cdot\frac{-2}{5}} \\ \implies x^{\frac{-5}{2}}=y \]
04

Simplify the expression

The expression is already simplified to \(y=x^{-\frac{5}{2}}\), and this is the formula for the inverse function. Therefore, the inverse function is \(f^{-1}(x) = x^{-\frac{5}{2}}\).

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Most popular questions from this chapter

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 \(s ?\)

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Show that if \(p\) and \(q\) are nonzero polynomials, then $$ \operatorname{deg}(p \circ q)=(\operatorname{deg} p)(\operatorname{deg} q) $$.

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