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Chapter 5: Problem 103

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

Short Answer

Expert verified
The cosine function must be restricted to the interval [0, \(\pi\)] so that it becomes one-to-one and therefore has an inverse function.

Step by step solution

01

Understand the Cosine Function

The cosine function, denoted by \(cos(x)\), is a periodic function that oscillates between -1 and 1 for every real number \(x\). The function repeats every \(2\pi\) intervals, creating a wave-like pattern. This means that there are infinitely many values of \(x\) that can give the same output, making the function not one-to-one.
02

Restriction of the Cosine Function

In order for a function to have an inverse, it must be one-to-one, i.e., each y-value has exactly one corresponding x-value. Since the cosine function is not one-to-one overall, we must restrict its domain to a portion where it is one-to-one. A commonly used restriction for the cosine function to make it one-to-one, is to limit its domain to \([0, \pi]\). Within this interval, each y-value corresponds to exactly one x-value, making the function one-to-one and thus has an inverse.
03

Conclusion

In conclusion, the cosine function, to have an inverse, must be restricted to a section of its domain where it is one-to-one. A commonly used restriction for this is \([0, \pi]\). This makes the cosine function one-to-one and thus allows it to have an inverse, which is referred to as the Inverse Cosine/Arc Cosine function, denoted by \(cos^{-1}(x)\) or \(arccos(x)\).

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