Problem 45
Question
In Exercises 43-48, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. \(h(x) = |x + 4| - |x-4|\)
Step-by-Step Solution
Verified Answer
Look at the graph obtained. Whether the function \(h(x) is one-to-one and has an inverse or not, completely depends upon the results of the Horizontal Line Test.
1Step 1 - Graph the Function
Use a graphing tool to create the graph of the function \(h(x) = |x + 4| - |x-4|\).
2Step 2 - Apply the Horizontal Line Test
Draw horizontal lines through the graph of the function. If any horizontal line intersects the graph in more than one place, the graph of the function doesn't pass the Horizontal Line Test. Therefore, the function isn't one-to-one and doesn't have an inverse.
3Step 3 - Interpret the Results
Based on the results of the Horizontal Line Test, if the test is passed (i.e., no horizontal line intersects the graph of the function in more than one place), then the function is one-to-one. Otherwise, it's not one-to-one.
Key Concepts
One-to-One FunctionInverse FunctionGraphing UtilityAbsolute Value Function
One-to-One Function
Understanding if a function is one-to-one is crucial for determining whether it has an inverse. A one-to-one function, also known as an injective function, is a type of function in which every element of the range is paired with exactly one element of the domain. To put it simply, no two different inputs in a one-to-one function will produce the same output.
An easy way to test if a function is one-to-one is by using the Horizontal Line Test. This can be done by graphing the function and then drawing horizontal lines across the graph. If any horizontal line intersects the function's graph more than once, then the function is not one-to-one. For our function, if any horizontal line touches the graph of the function at more than one point, we know that some outputs are repeated for different inputs, which violates the definition of a one-to-one function.
An easy way to test if a function is one-to-one is by using the Horizontal Line Test. This can be done by graphing the function and then drawing horizontal lines across the graph. If any horizontal line intersects the function's graph more than once, then the function is not one-to-one. For our function, if any horizontal line touches the graph of the function at more than one point, we know that some outputs are repeated for different inputs, which violates the definition of a one-to-one function.
Inverse Function
An inverse function essentially reverses the operation of the original function. If we have a function that takes a number and gives us an output, the inverse function would take that output and give us back the original number. However, for a function to have an inverse, it must be one-to-one as discussed before.
When we do have a one-to-one function, finding its inverse involves swapping the input and output values. In mathematical terms, if the function is denoted by f, and f(a) = b, then the inverse, denoted by f−1, would make f−1(b) = a. The inverse function graph is the reflection of the original graph across the line where y equals x, known as the line of symmetry. Using our function as an example, if it passes the Horizontal Line Test, we could calculate its inverse by solving the equation y = |x + 4| - |x - 4| for x.
When we do have a one-to-one function, finding its inverse involves swapping the input and output values. In mathematical terms, if the function is denoted by f, and f(a) = b, then the inverse, denoted by f−1, would make f−1(b) = a. The inverse function graph is the reflection of the original graph across the line where y equals x, known as the line of symmetry. Using our function as an example, if it passes the Horizontal Line Test, we could calculate its inverse by solving the equation y = |x + 4| - |x - 4| for x.
Graphing Utility
A graphing utility is a powerful tool that helps to visualize functions, which can be an essential aid in mathematics education and problem-solving. Such utilities range from graphing calculators, software programs, to online graphing apps. With technology, we can easily plot complex functions, perform the Horizontal Line Test, and analyze their characteristics.
When using a graphing utility to plot the function like our given absolute value function, we input the equation and then visually inspect the graph. The tool simplifies steps that would be meticulous and time-consuming to do by hand, such as checking for one-to-oneness, symmetry, or finding intercepts. Additionally, graphing utilities often come with features that let us adjust the viewing window or scale, so we can observe different behaviors of a function over varied intervals.
When using a graphing utility to plot the function like our given absolute value function, we input the equation and then visually inspect the graph. The tool simplifies steps that would be meticulous and time-consuming to do by hand, such as checking for one-to-oneness, symmetry, or finding intercepts. Additionally, graphing utilities often come with features that let us adjust the viewing window or scale, so we can observe different behaviors of a function over varied intervals.
Absolute Value Function
An absolute value function is written as f(x) = |x|, where the vertical bars represent the absolute value operation, which essentially strips a number of its sign. This means it turns all numbers positive. The graph of a simple absolute value function is a 'V' shape because all negative inputs are reflected over the y-axis to the positive side.
More complex absolute value functions, like the one in our exercise, can sometimes be split into piecewise functions to make graphing easier. For the function given, h(x) = |x + 4| - |x-4|, we are combining two different absolute value expressions, which makes its graph significantly different from the basic 'V'. It's also the reason why we need to graph the function to perform the Horizontal Line Test; such functions can behave unpredictably, and their one-to-one nature is not always obvious without visualization.
More complex absolute value functions, like the one in our exercise, can sometimes be split into piecewise functions to make graphing easier. For the function given, h(x) = |x + 4| - |x-4|, we are combining two different absolute value expressions, which makes its graph significantly different from the basic 'V'. It's also the reason why we need to graph the function to perform the Horizontal Line Test; such functions can behave unpredictably, and their one-to-one nature is not always obvious without visualization.
Other exercises in this chapter
Problem 44
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In Exercises 41-48, find (a) \(f \circ g\), and (b) \(g \circ f\). Find the domain of each function and each composite function. \(f(x) = |x|\), \(g(x) = x + 6\
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In Exercises 25-54, \(g\) is related to one of the parent functions described in Section 1.6. (a) Identify the parent function \(f\). (b) Describe the sequence
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