Question

the screen to the right shows how to define the inverse secant, cosecant, and cotangent functions in order to graph them using a ti - 83/84 plus graphing calculator.
use this information to graph the inverse circular function.
( y=cot ^{-1} 3 x )
choose the correct graph below
oa
ob
oc
plot 1 plot 2 plot 3
( y_1=cos ^{-1}(1 / x) )
( y_2=sin ^{-1}(1 / x) )
( y_3=\tan ^{-1}(1 / x) )
( y_4=+pi) *(x<0)+\tan ^{-1}(1 / x) )
( y_5= )

Explanation:

Step1: Recall the properties of \(y = \cot^{-1}(x)\)

The function \(y=\cot^{-1}(x)\) has a domain of \((-\infty,\infty)\) and a range of \((0,\pi)\). It is a decreasing function. For \(y = \cot^{-1}(3x)\), the horizontal compression factor is \(a = 3\). The general form of the inverse - cotangent function \(y=\cot^{-1}(ax)\) still has a range of \((0,\pi)\) and is a decreasing function.

Step2: Analyze the options

• Option A: The graph of \(y = \cot^{-1}(3x)\) is a decreasing function. As \(x\to-\infty\), \(y\to\pi\) and as \(x\to\infty\), \(y\to0\). This option matches the behavior of the inverse - cotangent function \(y=\cot^{-1}(3x)\) (a decreasing function).
• Option B: The graph in option B does not match the range and decreasing nature of \(y=\cot^{-1}(3x)\) (the range of \(y = \cot^{-1}(3x)\) is \((0,\pi)\) and it is a decreasing function, while the graph in option B has an incorrect orientation).
• Option C: The graph in option C is an increasing function, which does not match the property of \(y=\cot^{-1}(3x)\) (since \(y = \cot^{-1}(u)\) is a decreasing function of \(u\) and \(u = 3x\) is a linear function, and by the composition of functions, \(y=\cot^{-1}(3x)\) is a decreasing function).

Answer:

A.