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Chapter 3: Problem 54

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=5^{x}-3 $$

Short Answer

Expert verified
The inverse function \(f^{-1}(x)\) of the given function \(f(x) = 5^x - 3\) is: $$ f^{-1}(x) = \log_5{(x+3)} $$

Step by step solution

01

Rewrite the given function

First, write the given function using the notation y instead of f(x): $$ y = 5^x - 3 $$
02

Solve for x in terms of y

Next, we need to isolate x on one side of the equation. To do this, follow these steps: 1. Add 3 to both sides of the equation to get rid of the constant term: $$ y + 3 = 5^x $$ 2. Now we need to get rid of the exponent by taking the logarithm of both sides. We can use the logarithm with base 5 (written as \(\log_5\) or \(lg_5\)) for simplicity: $$ \log_5{(y+3)} = \log_5{(5^x)} $$ 3. Using the logarithm property, we can bring the exponent, x, down in front of the logarithm: $$ \log_5{(y+3)} = x\log_5{5} $$ 4. Since \(\log_5{5} = 1\), we get: $$ x = \log_5{(y+3)} $$
03

Switch the roles of x and y

Now, to find the formula for the inverse function \(f^{-1}(x)\), we switch the roles of x and y: $$ y = \log_5{(x+3)} $$
04

Write the formula for the inverse function

Finally, we have the formula for the inverse function \(f^{-1}(x)\): $$ f^{-1}(x) = \log_5{(x+3)} $$

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Most popular questions from this chapter

Explain why $$ \ln x \approx 2.302585 \log x $$ for every positive number \(x\).

Estimate the indicated value without using a calculator. \(\left(\frac{e^{7.001}}{e^{7}}\right)^{2}\)

Suppose \(f\) is the function defined by $$ f(x)=\cosh x $$ for every \(x \geq 0\). In other words, \(f\) is defined by the same formula as cosh, but the domain of \(f\) is the interval \([0, \infty)\) and the domain of cosh is the set of real numbers. Show that \(f\) is a one-to-one function and that its inverse is given by the formula $$ f^{-1}(y)=\ln \left(y+\sqrt{y^{2}-1}\right) $$ for every \(y \geq 1\).

Find a number \(y\) such that \(e^{4 y-3}=5\).

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