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Chapter 4: Problem 62

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=-6 x-8$$

Short Answer

Expert verified
(a) \(f^{-1}(x) = -\frac{x + 8}{6}\) (b) Reflect \(y = -6x-8\) over \(y = x\) (c) Domain and range are both \(\mathbb{R}\).

Step by step solution

01

Verify One-to-One Nature

A function is one-to-one if it passes the horizontal line test or if its derivative does not change sign. For the linear function \(f(x) = -6x - 8\), which has the form \(y = mx + b\), the slope \(m\) is constant and nonzero. Therefore, \(f(x) = -6x - 8\) is one-to-one.
02

Find the Inverse Function

To find the inverse function \(f^{-1}(x)\), start by replacing \(f(x)\) with \(y\):
03

Step 2.1: Swap Variables

Write the equation \(y = -6x - 8\), then swap \(x\) and \(y\) to get \(x = -6y - 8\).
04

Step 2.2: Solve for y

Now, solve for \(y\): \[x = -6y - 8\] Add 8 to both sides: \[x + 8 = -6y\] Divide both sides by -6: \[y = -\frac{x + 8}{6}\] So, \(f^{-1}(x) = -\frac{x + 8}{6}\).
05

Graph the Functions

Graph \(f(x) = -6x - 8\) and its inverse \(f^{-1}(x) = -\frac{x + 8}{6}\) on the same axes, making sure to reflect the original function over the line \(y = x\). The graph of the inverse will be a reflection of the graph of the original function.
06

Determine the Domain and Range

For the function \(f(x) = -6x - 8\), any value of \(x\) is allowed, so the domain is all real numbers, \(\mathbb{R}\). Since \(f(x)\) can take any real value, its range is also \(\mathbb{R}\).For the inverse function \(f^{-1}(x) = -\frac{x + 8}{6}\), the domain and range are also all real numbers, \(\mathbb{R}\).

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Key Concepts

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

One-to-One Functions
A one-to-one function is a type of function where each output value corresponds to exactly one input value. This means every element in the domain maps to a unique element in the range. To determine if a function is one-to-one, we can use different methods:
  • The Horizontal Line Test: A function is one-to-one if no horizontal line intersects its graph more than once.
  • The Derivative Test: If the derivative of the function does not change sign (it is always positive or always negative), the function is one-to-one.
Let鈥檚 consider the function from our exercise: \(f(x) = -6x - 8\). Since this is a linear function with a non-zero slope of -6, it is one-to-one. The constant non-zero slope indicates that it consistently increases or decreases, passing the horizontal line test. Hence, we can conclude that \(f(x) = -6x - 8\) is a one-to-one function.
Domain and Range
The domain of a function consists of all the possible input values (x-values) for which the function is defined. The range is the set of all possible outputs (y-values) that the function can produce.
For the linear function \(f(x) = -6x - 8\):
  • Domain: Since it is defined for every real number, the domain is \(\mathbb{R}\)
  • Range: Because the function can output any real number as \(x\) takes any value, the range is also \(\mathbb{R}\)
For the inverse function \(f^{-1}(x) = -\frac{x + 8}{6}\):
  • Domain: The domain of the inverse function corresponds to the range of the original function, which is \(\mathbb{R}\)
  • Range: The range of the inverse function corresponds to the domain of the original function, which is also \(\mathbb{R}\)
Therefore, both the domain and the range of \(f(x)\) and \(f^{-1}(x)\) are all real numbers.
Graphing Functions
Graphing functions helps us visualize their behavior. For one-to-one functions, it's also important for understanding the relationship between a function and its inverse. Let鈥檚 graph both \(f(x) = -6x - 8\) and its inverse \(f^{-1}(x) = -\frac{x + 8}{6}\) on the same set of axes:
  • First, plot the original function \(f(x) = -6x - 8\).
  • Then, plot the inverse function \(f^{-1}(x) = -\frac{x + 8}{6}\), which is a reflection of the original function over the line \(y = x\).
The reflection property means that if \( (a, b)\) lies on the graph of \(f(x)\), then \( (b, a)\) will lie on the graph of \(f^{-1}(x)\). By graphing these functions, it becomes clear how each point on \(f(x)\) corresponds to its reflected point on \(f^{-1}(x)\), illustrating the one-to-one relationship and the inversion process effectively.

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Most popular questions from this chapter

(Modeling) Solve each problem. See Example 11 . Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table. $$\begin{array}{c|c||c|c} \hline \text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\ \hline 0 & 1013 & 6000 & 472 \\ \hline 1000 & 899 & 7000 & 411 \\ \hline 2000 & 795 & 8000 & 357 \\ \hline 3000 & 701 & 9000 & 308 \\ \hline 4000 & 617 & 10,000 & 265 \\ \hline 5000 & 541 & & \\ \hline \end{array}$$ (a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure \(P\) at altitude \(x\). (b) Would a linear or an exponential function fit the data better? (c) The function $$ P(x)=1013 e^{-0.0001341 x} $$ approximates the data. Use a graphing calculator to graph \(P\) and the data on the same coordinate axes. (d) Use \(P\) to predict the pressures at \(1500 \mathrm{m}\) and \(11,000 \mathrm{m},\) and compare them to the actual values of 846 millibars and 227 millibars, respectively.

For individual or collaborative investigation (Exercises \(117-122\) ) Assume \(f(x)=a^{x}\), where \(a>1 .\) Work these exercises in order. If \(\left.a=10, \text { what is the equation for } y=f^{-1}(x) ? \text { (You need not solve for } y .\right)\)

Emissions Tax One action that government could take to reduce carbon emissions into the atmosphere is to levy a tax on fossil fuel. This tax would be based on the amount of carbon dioxide emitted into the air when the fuel is burned. The cost-benefit equation $$\ln (1-P)=-0.0034-0.0053 T$$ models the approximate relationship between a tax of \(T\) dollars per ton of carbon and the corresponding percent reduction \(P\) (in decimal form) of emissions of carbon dioxide. (Source: Nordhause, W., "To Slow or Not to Slow: The Economics of the Greenhouse Effect," Yale University, New Haven, Connecticut.) (a) Write \(P\) as a function of \(T\). (b) Graph \(P\) for \(0 \leq T \leq 1000 .\) Discuss the benefit of continuing to raise taxes on carbon (c) Determine \(P\) when \(T=60\) dollars, and interpret this result. (d) What value of \(T\) will give a \(50 \%\) reduction in carbon emissions?

If the function is one-to-one, find its inverse. $$\left\\{(3,-1),(5,0),(0,5),\left(4, \frac{2}{3}\right)\right\\}$$

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=\frac{1}{x-3}, \quad x \neq 3$$
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