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Chapter 4: Problem 103

Describe the restriction on the cosine function so that it has an inverse function.

Short Answer

Expert verified
The cosine function has an inverse when it's restricted to an interval where it's either strictly increasing or decreasing. One common choice is the interval [0, \(\pi\)].

Step by step solution

01

Define the cosine function and its properties

The cosine function, cos(x), oscillates between -1 and 1 for all real numbers x. It reaches its maximum value of 1 at x = 0 and its minimum value of -1 at x = \(\pi\). Since these maxima and minima happen periodically every 2\(\pi\) units, the function repeats itself every 2\(\pi\) units.
02

Apply the horizontal line test

The horizontal line test states that a function f has an inverse if and only if no horizontal line intersects the graph of f at more than one point. Since cosine reaches both its maxima and minima infinitely many times, the graph of cos(x) does not pass this test when considering all real numbers x. Therefore, cos(x) does not have an inverse over its entire domain.
03

Restrict the domain of the cosine function

To make cos(x) have an inverse, we need to restrict its domain to an interval where it's either strictly increasing or decreasing. One such interval is [0, \(\pi\)]. Within this interval, the cosine function decreases from 1 to -1, which means it passes the horizontal line test.
04

Verify the inverse

When the cosine function is restricted to [0, \(\pi\)], it has an inverse function, which is typically written as cos^-1(x) or arccos(x). This function gives the angle (in the range [0, \(\pi\)]) whose cosine is x.

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Most popular questions from this chapter

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