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Chapter 8: Problem 71

What is the horizontal line test and what does it indicate?

Short Answer

Expert verified
The horizontal line test is a technique used to check if a function is one-to-one (injective). It involves drawing a horizontal line through the graph of the function. If the line touches the graph at more than one point, the function is not one-to-one; hence it does not have an exact inverse. This test is significant as it helps to verify the existence of an inverse function.

Step by step solution

01

Define the Horizontal line test

The horizontal line test is a way to determine if a function is injective (or one-to-one). It's performed by drawing a horizontal line that passes through the graph of the function. If the horizontal line touches the graph in more than one point, then the function is not one-to-one.
02

Explain the Importance

The importance of the horizontal line test lies in its ability to tell us whether a function has an inverse. For a precise inverse of a function to exist, every input must correspond to exactly one output and every output must correspond to exactly one input. This cannot be the case if the horizontal line passes through more than one point on the graph of the function, because this means that more than one input (where the horizontal line intersects the graph) corresponds to a single output (the y-coordinate of the line).
03

Provide Visual Representation

Visualizing the concept would help to understand it better. Consider trying to draw a graph of a function and a horizontal line that goes through it. If the horizontal line intersects the graph at more than one point, it means the function isn't one-to-one and doesn't have an exact inverse.

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Most popular questions from this chapter

Use a graphing utility to graph each function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$f(x)=\sqrt[3]{2-x}$$

Use a graphing utility to graph each function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$f(x)=\frac{x^{4}}{4}$$

Will help you prepare for the material covered in the first section of the next chapter. Solve: \(2-12 x=7(x-1)\).

Find a. \((f \circ g)(x)\), b. \((g \circ f)(x)\), c. \((f \circ g)(2)\). $$f(x)=7 x+1, \quad g(x)=2 x^{2}-9$$

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=\frac{2 x-3}{x+1}$$
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