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Chapter 1: Problem 42

Write the inverse for each function. $$ s(v)=e^{v}, v>0 $$

Short Answer

Expert verified
The inverse function is \( s^{-1}(y) = \ln(y) \) for \( y > 1 \).

Step by step solution

01

Understand the Function

The given function is \( s(v) = e^v \), where \( v > 0 \). This is an exponential function with the base \( e \). Our goal is to find the inverse of this function.
02

Define the Inverse Function

To find the inverse function, we start by switching the roles of the input and the output. Set \( y = s(v) \), then \( y = e^v \). We need to express \( v \) in terms of \( y \).
03

Solve for v

Take the natural logarithm of both sides of the equation \( y = e^v \) to solve for \( v \). This gives us: \( \ln(y) = \ln(e^v) \). The property of logarithms \( \ln(e^v) = v \) allows us to simplify to \( v = \ln(y) \).
04

Write the Inverse Function

Replace \( y \) with the function notation for the inverse. The inverse function of \( s(v) \) is \( s^{-1}(y) = \ln(y) \) for \( y > 1 \) since \( s(v) = e^v \) and \( v > 0 \).
05

Check the Domain and Range

Since \( v > 0 \), \( e^v \) is always greater than 1. Thus, the domain of the inverse function \( s^{-1}(y) \) is \( y > 1 \). The range of the inverse function is \( v > 0 \), the same as the domain of the original function.

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Key Concepts

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

Exponential Functions
Exponential functions are a special type of mathematical function where the variable is in the exponent. In our exercise, the function is represented as \( s(v) = e^v \) where \( e \) is the base of the natural logarithm, approximately equal to 2.718. Exponential functions have the form \( f(x) = a^x \) where \( a \) is a positive constant, and \( x \) is the variable. They are widely used in various fields such as finance, physics, and biology to model continuous growth or decay.
  • The base \( e \) is natural because it arises naturally in the process of continuous growth.
  • Exponential functions grow rapidly as the variable increases, leading to a characteristic curve that sharply rises.
  • In an exponential function like \( e^v \), the growth rate is proportional to its value at any point.
Understanding exponential functions is crucial because they often involve transformations that require inversion to analyze or satisfy given conditions.
Logarithms
Logarithms are the inverse operations of exponentiation. If you have an exponential equation like \( y = e^v \), finding the corresponding \( v \) is achieved through logarithms. The natural logarithm, denoted \( \ln(x) \), is specifically used when the base of the exponent is \( e \). In our solution, we find the inverse by applying the natural logarithm: \( v = \ln(y) \).
  • The natural logarithm function \( \ln(x) \) is defined only for \( x > 0 \).
  • Logarithms turn multiplication into addition, which simplifies calculations in many practical scenarios.
  • Key property: \( \ln(e^x) = x \), because the exponential and logarithmic functions are inverses.
Thus, logarithms help us manipulate and solve equations involving exponential functions effectively.
Domain and Range
The domain and range of a function define the set of possible inputs (domain) and outputs (range). For the function \( s(v) = e^v \), the domain is \( v > 0 \) because the exponent \( v \) is specified as greater than zero. This means \( e^v \) produces values greater than 1, so the range of \( s(v) \) is also \( y > 1 \).
When considering the inverse function \( s^{-1}(y) = \ln(y) \), the domain becomes \( y > 1 \) (which matches the range of \( s(v) \)), and the range is \( v > 0 \), aligning with the original function's domain.
  • The domain of a function is crucial as it defines the set of valid inputs you can work with.
  • The range describes all potential outputs that the function can generate.
  • For inverse functions, recognizing domain and range ensures they properly map back to the original function.
Understanding domain and range helps in graphing functions and ensuring any mathematical operations and transformations remain valid.

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Most popular questions from this chapter

a. Describe the behavior suggested by a scatter plot of the data and list the types of models that exhibit this behavior. b. Describe the possible end behavior as input increases and list the types of models that would fit each possibility. c. Write the model that best fits the data. d. Write the model that best exhibits the end behavior of the data. Production, Given the Amount Invested in Capital$$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Capital } \\ \text { (million dollars) } \end{array} & \begin{array}{c} \text { Production } \\ \text { (billion units) } \end{array} \\ \hline 6 & 19 \\ \hline 18 & 38 \\ \hline 24 & 42 \\ \hline 30 & 45 \\ \hline 42 & 60 \\ \hline 48 & 77 \\ \hline \end{array} $$

Rewrite each pair of functions as one composite function and evaluate the composite function at 2. $$ h(p)=\frac{4}{p} ; p(t)=1+3 e^{-0.5 t} $$

On the Arctic Circle, there are 24 hours of daylight on the summer solstice, June 21 (the 173rd day of the year), and 24 hours of darkness on the winter solstice, December 21 (day -10 and day 355 ). There are 12 hours of daylight and 12 hours of darkness midway between the summer and winter solstices (days 81 and 264 ). a. What are the maximum and minimum hours of daylight in the Arctic Circle? Use these values to calculate the amplitude and average value of the hours of daylight cycle. b. Calculate the period and horizontal shift of the hours of daylight cycle. Use these values to calculate the parameters \(b\) and \(c\) for a model of the form \(f(x)=a \sin (b x+c)+d\) c. Use the constants from parts \(a\) and \(b\) to construct a sine model for the hours of daylight in the Arctic Circle.

Braking Distance The table below gives the results of an online calculator showing how far (in feet) a vehicle will travel while braking to a complete stop, given the initial velocity of the automobile. Braking Distance $$ \begin{array}{c|c} \text { MPH } & \text { Distance (feet) } \\ \hline 10 & 27 \\ \hline 20 & 63 \\ \hline 30 & 109 \\ \hline 40 & 164 \\ \hline 50 & 229 \\ \hline 60 & 304 \\ \hline 70 & 388 \\ \hline 80 & 481 \\ \hline 90 & 584 \\ \hline \end{array} $$ a. Find a quadratic model for the stopping distance. b. What other factors besides the initial speed would affect the stopping distance?

Stolen Bases San Francisco Giants legend Willie Mays's cumulative numbers of stolen bases between 1951 and 1963 are as shown below. Bases Stolen by Willie Mays (cumulative) \begin{tabular}{|c|c|c|c|} \hline Year & Stolen Bases & Year & Stolen Bases \\ \hline 1951 & 7 & 1958 & 152 \\ \hline 1952 & 11 & 1959 & 179 \\ \hline 1953 & 11 & 1960 & 204 \\ \hline 1954 & 19 & 1961 & 222 \\ \hline 1955 & 43 & 1962 & 240 \\ \hline 1956 & 83 & 1963 & 248 \\ \hline 1957 & 121 & & \\ \hline \end{tabular} (Source: www.baschall-reference.com) a. Find a logistic model for the data with input data aligned so that \(t=0\) in \(1950 .\) b. According to the model, how many bases did Mays steal in 1964 ? c. In 1964 Mays stole 19 bases. Does the model overestimate or underestimate the actual number? By how much?
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