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Chapter 1: Problem 44

Use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function.$$h(r)=|r+4|-|r-4|$$

Short Answer

Expert verified
The answer is dependent on the results of the Horizontal Line Test as applied in Step 2 of the solution.

Step by step solution

01

Graphing the function

Begin by sketching the graph of the function \(h(r) = |r + 4| - |r - 4|\) using a graphing utility. The graph is essential as it allows to check for the function's invertibility using the Horizontal Line Test.
02

Applying the Horizontal Line Test

Once the graph is formed, proceed to the Horizontal Line Test. It involves taking a horizontal line and running it through the graph. If any horizontal line crosses the function more than once, the function does not pass the test and thus does not have an inverse. If each horizontal line intersects the function at most once, then the function is valid to have an inverse.
03

Determining the existence of an inverse

Following the results of Step 2's line test, if the function passes, it can be concluded that the function has an inverse. If it does not pass, however, an inverse does not exist for the given function.

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Key Concepts

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

Horizontal Line Test
The Horizontal Line Test helps determine if a function has an inverse. It is a visual way of checking invertibility by using horizontal lines across the graph of a function.
  • If a horizontal line intersects the graph at more than one point, the function fails the test. This indicates that there are multiple outputs for a single input, which means the function does not have an inverse.
  • If each horizontal line intersects the graph at only one point, the function passes the test and has an inverse. This means that each output corresponds to one unique input, confirming that the function is one-to-one.
For the exercise given, after graphing the function with a graphing utility, it was determined using the Horizontal Line Test that the function did not pass, indicating no inverse function exists. The test offers a simple method to visualize and understand the concept of function invertibility.
graphing utility
A graphing utility is a technological tool used to sketch and analyze graphs of functions. These devices or apps allow us to input mathematical functions and obtain their graphical representation very efficiently.
For the function \(h(r) = |r+4| - |r-4|\), a graphing utility was employed to visualize how the function behaves. This function has two absolute value expressions, making it more complex to graph manually.
Graphing utilities simplify these tasks by:
  • Providing accurate and quick representation of the mathematical functions.
  • Allowing manipulation of the view to better examine parts of the graph.
  • Offering tools to apply tests such as the Horizontal Line Test.
By using these utilities, such analyses become accessible to students regardless of their manual graphing skills, helping them understand the properties of functions.
absolute value function
An absolute value function involves expressions within absolute value signs, typically denoted as \(|x|\). It measures how far a number is from zero on the number line, regardless of direction.
The function \(h(r) = |r+4| - |r-4|\) contains two absolute value terms. Here鈥檚 how they work:
  • \(|r+4|\) measures the distance of \(r\) from \(-4\).
  • \(|r-4|\) measures the distance of \(r\) from \(4\).
When these two expressions are used in a single function, the resultant graph can change its nature at any points where the expressions inside the absolute value signs change from positive to negative. This makes the function more dynamic and influences whether it passes the Horizontal Line Test, as seen in this exercise.
invertibility
Invertibility is a critical property of functions that determines whether a function has an inverse. If a function has an inverse, each input is uniquely paired with an output.
For a function to be invertible:
  • It must pass the Horizontal Line Test, indicating it is one-to-one (bijective).
  • Each \(y\) value in the range must correspond to exactly one \(x\) value in the domain.
In the given solution, the graph of \(h(r) = |r+4| - |r-4|\) was analyzed and failed the Horizontal Line Test. Therefore, this function is not invertible. Understanding invertibility is important not just for graph analysis, but also for solving real-world problems where reverse calculations are necessary.

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Most popular questions from this chapter

The function $$y=0.03 x^{2}+245.50, \quad 0< x <100$$,approximates the exhaust temperature \(y\) in degrees Fahrenheit, where \(x\) is the percent load for a diesel engine. (a) Find the inverse function. What does each variable represent in the inverse function? (b) Use a graphing utility to graph the inverse function. (c) The exhaust temperature of the engine must not exceed 500 degrees Fahrenheit. What is the percent load interval?

(a) Given a function \(f,\) prove that \(g(x)\) is even and \(h(x)\) is odd, where \(g(x)=\frac{1}{2}[f(x)+f(-x)]\) and \(h(x)=\frac{1}{2}[f(x)-f(-x)].\) (b) Use the result of part (a) to prove that any function can be written as a sum of even and odd functions. [Hint: Add the two equations in part (a).] (c) Use the result of part (b) to write each function as a sum of even and odd functions. \(f(x)=x^{2}-2 x+1, \quad k(x)=\frac{1}{x+1}\)

Find a mathematical model for the verbal statement. \(F\) varies directly as \(g\) and inversely as \(r^{2}\).

The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the driver's reaction time is given by \(R(x)=\frac{3}{4} x,\) where \(x\) is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given by \(B(x)=\frac{1}{15} x^{2}.\) (a) Find the function that represents the total stopping distance \(T.\) (b) Graph the functions \(R, B,\) and \(T\) on the same set of coordinate axes for \(0 \leq x \leq 60.\) (c) Which function contributes most to the magnitude of the sum at higher speeds? Explain.

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\frac{6 x+4}{4 x+5}$$
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