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Chapter 3: Problem 64

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=8+9 \log _{2}(4 x-7) $$

Short Answer

Expert verified
The short answer for the inverse function, \(f^{-1}(x)\), of the given function is: \(f^{-1}(x) = \frac{2^{\frac{x - 8}{9}}+7}{4}\).

Step by step solution

01

Write down the given function and replace the function notation with inverse function notation.

Let's write down the given function: $$ f(x) = 8 + 9\log_2{(4x - 7)} $$ Now, replace \(f(x)\) with \(f^{-1}(y)\) and \(x\) with \(y\): $$ f^{-1}(y) = 8 + 9\log_2{(4y - 7)} $$
02

Rearrange the equation to isolate the logarithmic term.

To isolate the logarithmic term, subtract 8 from both sides: $$ f^{-1}(y) - 8 = 9\log_2{(4y - 7)} $$
03

Divide both sides by the coefficient of the logarithmic term.

Divide both sides of the equation by 9: $$ \frac{f^{-1}(y) - 8}{9} = \log_2{(4y - 7)} $$
04

Use the property of logarithms to rewrite the logarithmic equation as an exponential equation.

Using the property of logarithms, we can rewrite the logarithmic equation as an exponential equation: $$ 2^{\frac{f^{-1}(y) - 8}{9}} = 4y - 7 $$
05

Solve for y.

To solve for \(y\), first isolate the term \(4y\): $$ 2^{\frac{f^{-1}(y) - 8}{9}} + 7 = 4y $$ Now, divide both sides by 4: $$ y = \frac{2^{\frac{f^{-1}(y) - 8}{9}}+7}{4} $$
06

Replace y with the inverse function notation.

Replace \(y\) with \(f^{-1}(x)\): $$ f^{-1}(x) = \frac{2^{\frac{x - 8}{9}}+7}{4} $$ The inverse function of the given function is: $$ f^{-1}(x) = \frac{2^{\frac{x - 8}{9}}+7}{4} $$

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Most popular questions from this chapter

Show that \(\cosh x \geq 1\) for every real number \(x\).

Estimate the given number. Your calculator will be unable to evaluate directly the expressions in these exercises. Thus you will need to do more than button pushing for these exercises. \(\left(1-\frac{4}{9^{80}}\right)^{\left(9^{80}\right)}\)

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=-6+7 \ln x\)

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=4-2 e^{8 x}\)

Find all numbers \(x\) that satisfy the given equation. \(\ln (x+5)-\ln (x-1)=2\)
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