Use the horizontal line test to determine whether the...

Name: Problem 34, Chapter 3: Precalculus
Uploaded: 2020-07-30T00:50:10-08:00
Duration: 1 min 36 s
Channel: Bobby Barnes
Description: Use the horizontal line test to determine whether the function is one-to-one (and therefore has an inverse ). (You should be able to sketch the graph of each function on your own, without using a graphing utility.) Which of the six basic functions graphed in Figure 7 in Section 3.2 are one-to-one?

Use the horizontal line test to determine whether the function is one-to-one (and therefore has an inverse ). (You should be able to sketch the graph of each function on your own, without using a graphing utility.) Which of the six basic functions graphed in Figure 7 in Section 3.2 are one-to-one?

Use the horizontal line test to determine whether the function is one-to-one (and therefore has an inverse ). (You should be able to sketch the graph of each function on your own, without using a graphing utility.)
Which of the six basic functions graphed in Figure 7 in Section 3.2 are one-to-one?

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Key Concepts

Basic Functions and Graphical Analysis

Understanding the characteristics of basic functions, including their standard shapes and properties, is essential for sketching their graphs and applying the horizontal line test. Familiarity with these basic functions enables one to quickly identify which functions are one-to-one and can, therefore, have an inverse.

Inverse Function

An inverse function essentially reverses the original function, mapping outputs back to their corresponding inputs. The existence of an inverse function requires that the original function be one-to-one, ensuring that every output is uniquely produced by a single input.

One-to-One Function

A one-to-one function, or injective function, is defined such that each element of the function's domain maps to a unique element in its range. This property is critical because only one-to-one functions can have inverses; otherwise, the reverse mapping would be ambiguous.

Horizontal Line Test

The horizontal line test is a visual method used to determine if a function is one-to-one. By drawing horizontal lines across the graph of the function, one can check whether any horizontal line intersects the graph more than once. If it does, the function does not have a unique inverse since a single output would correspond to multiple inputs.

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