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Chapter 4: Problem 87

Find an equation for the inverse function. \(f(x)=\ln (x+5)\)

Short Answer

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

Step by step solution

01

Understand the Function

The given function is the natural logarithm function: \(f(x) = \ln (x+5)\). To find its inverse, we need to express the original function in terms of \(y\).
02

Replace \(f(x)\) with \(y\)

First, replace \(f(x)\) with \(y\). We get: \(y = \ln (x+5)\).
03

Swap \(x\) and \(y\)

The next step in finding the inverse function is to swap \(x\) and \(y\). This gives us: \(x = \ln (y+5)\).
04

Solve for \(y\)

To isolate \(y\), exponentiate both sides to remove the natural logarithm. This gives us: \ e^x = y+5\.
05

Isolate \(y\)

Finally, solve for \(y\) by subtracting 5 from both sides: \(y = e^x - 5\). Therefore, the inverse function \(f^{-1}(x)\) is \(e^x - 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
A natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. It is the inverse operation of exponentiation with base \(e\). This means if you have \(e^y = x\), then \(\ln(x) = y\).

For example, if you have \(\ln(x) = 2\), it implies that \(x\) is equal to \(e^2\), which is roughly 7.389. The natural logarithm function is widely used in fields like mathematics, physics, engineering, and economics due to its unique properties related to growth processes.
Exponentiation
Exponentiation is an arithmetic operation that involves raising a number, known as the base, to the power of an exponent. It is written as \(a^n\), where \(a\) is the base and \(n\) is the exponent. The result represents the base multiplied by itself \(n\) times.

In the context of the natural logarithm, exponentiation is used to 'undo' the logarithmic function. For instance, if you have a logarithmic equation \(y = \ln (x+5)\), you exponentiate both sides to solve for \(x\), giving \(e^y = x+5\). This transformation is crucial in finding the inverse function, as demonstrated in the solution to the problem at hand.
Function Transformation
Function transformation involves altering the appearance or position of a graph. There are several types of function transformations, including shifts, reflections, stretches, and compressions.

For the function \(f(x) = \ln (x+5)\), we observe a horizontal shift to the left by 5 units. This is because the argument inside the logarithm, \(x+5\), moves the graph of the basic function \(\ln(x)\) accordingly.

Understanding such transformations is essential when dealing with functions and their inverses, as it helps visualize how the graph of the inverse function relates to the original.
Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is the language through which we describe patterns and relationships.

In solving for the inverse function of \(f(x) = \ln (x+5)\), algebraic manipulation is key. We start with the equation \(y = \ln (x+5)\) and use algebraic operations such as swapping variables and exponentiation to isolate \(y\). This involves:
  • Replacing \(f(x)\) with \(y\)
  • Swapping \(x\) and \(y\)
  • Exponentiating both sides to remove the \(\ln\)
  • Solving for \(y\)
So, through these algebraic steps, we determine the inverse function as \(f^{-1}(x) = e^x - 5\).

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Most popular questions from this chapter

Painful bone metastases are common in advanced prostate cancer. Physicians often order treatment with strontium- \(89\left({ }^{89} \mathrm{Sr}\right)\), a radionuclide with a strong affinity for bone tissue. A patient is given a sample containing \(4 \mathrm{mCi}\) of \({ }^{89} \mathrm{Sr}\). a. If \(20 \%\) of the \({ }^{89} \mathrm{Sr}\) remains in the body after 90 days, write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the amount \(Q(t)\) of radioactivity in the body \(t\) days after the initial dose. b. What is the biological half-life of \({ }^{89} \mathrm{Sr}\) under this treatment?

Use the model \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) The variable A represents the future value of P dollars invested at an interest rate \(r\) compounded \(n\) times per year for \(t\) years. If 4000 is put aside in a money market account with interest reinvested monthly at 2.2%, find the time required for the account to earn 1000. Round to the nearest month.

Use the model \(A=P e^{r t} .\) The variable \(A\) represents the future value of \(P\) dollars invested at an interest rate \(r\) compounded continuously for \(t\) years. If a couple has \(\$ 80,000\) in a retirement account, how long will it take the money to grow to \(\$ 1,000,000\) if it grows by \(6 \%\) compounded continuously? Round to the nearest year.

a. Graph \(f(x)=\ln x\) and \(g(x)=(x-1)-\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}-\frac{(x-1)^{4}}{4}\) on the viewing window [-2,4,1] by [-5,2,1] . How do the graphs compare on the interval (0,2) ? b. Use function \(g\) to approximate \(\ln 1.5\). Round to 4 decimal places.

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 3 & 2.7 \\ \hline 7 & 12.2 \\ \hline 13 & 25.7 \\ \hline 15 & 30 \\ \hline 17 & 34 \\ \hline 21 & 44.4 \\ \hline \end{array} $$
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