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Chapter 4: Problem 115

Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi),\) and sketch the graph of the inverse trigonometric function.

Short Answer

Expert verified
The inverse cotangent function, cot鈦宦�(x), is defined by restricting the domain of the cotangent function to the interval (0, 蟺). Its graph starts from 蟺 at x=0 and decreases to 0 as x tends to 鈭�, never touching the x-axis.

Step by step solution

01

Understand the cotangent function

The cotangent function, generally written as cot(x), is the reciprocal of the tangent function, or cot(x)=1/tan(x). It is periodic and repeats every 蟺. It is undefined wherever tangent is 0.
02

Restrict the domain of the cotangent function

The domain of the original cotangent function is all real numbers except multiples of 蟺, where the function is undefined. To find the inverse of the cotangent function, we restrict the domain to (0, 蟺). This is because in this interval, for every output there is exactly one input, so we can then create an inverse function.
03

Define the inverse cotangent function

The restricted cotangent function will yield the inverse cotangent function, sometimes written as arccot, or cot鈦宦�(x). If y = cot鈦宦�(x), this means that cot(y) = x for y in (0, 蟺). The range of the inverse cotangent function is, as mentioned, the interval (0, 蟺).
04

Sketch the graph of the inverse cotangent function

The graph of y = cot鈦宦�(x) is derived from the cotangent graph, but it's reflected around the line y=x, because this is how any inverse function is graphed. In the first quadrant, the function starts from 蟺 at x=0 and decreases continuously, reaching 0 at x = 鈭�. The graph never actually touches the x-axis.

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Most popular questions from this chapter

Find the exact value of the expression. (Hint: Sketch a right triangle.) \(\sec \left[\arctan \left(-\frac{3}{5}\right)\right]\)

Consider the functions \(f(x)=\sin x\) and \(f^{-1}(x)=\arcsin x\). (a) Use a graphing utility to graph the composite functions \(f \circ f^{-1}\) and \(f^{-1} \circ f\). (b) Explain why the graphs in part (a) are not the graph of the line \(y=x .\) Why do the graphs of \(f \circ f^{-1}\) and \(f^{-1} \circ f\) differ?

Use the properties of inverse trigonometric functions to evaluate the expression. \(\sin (\arcsin 0.3)\)

Use a graphing utility to graph \(f\) \(g,\) and \(y=x\) in the same viewing window to verify geometrically that \(g\) is the inverse function of \(f .\) (Be sure to restrict the domain of \(f\) properly.) \(f(x)=\cos x, \quad g(x)=\arccos x\)

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window to verify that the two functions are equal. Explain why they are equal. Identify any asymptotes of the graphs. \(f(x)=\tan \left(\arccos \frac{x}{2}\right), \quad g(x)=\frac{\sqrt{4-x^{2}}}{x}\)
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