Back to Tutor Intake Assessment: Restrict domain for inverse trig functions

HSF.TF.B.6 Tutor Intake -- Inverse Trig Domain Restrictions

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Grade 9·11 problems·~14 min·Common Core Math - HS Functions·group·hsf-tf-b-6

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Concepts

Why does $y = \sin(x)$ not have an inverse function on its full domain?

It fails the horizontal line test — many $x$ -values share the same $y$ -value.

It is not defined for all real numbers.

Its range does not include all real numbers.

It passes the vertical line test.

Which row correctly lists all three standard restricted domains used to
define $\arcsin$ , $\arccos$ , and $\arctan$ ?

$\arcsin$ : $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$ ;
$\arccos$ : $[0, \pi]$ ;
$\arctan$ : $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$

$\arcsin$ : $[0, \pi]$ ;
$\arccos$ : $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$ ;
$\arctan$ : $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$

$\arcsin$ : $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$ ;
$\arccos$ : $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$ ;
$\arctan$ : $(0, \pi)$

$\arcsin$ : $[-1, 1]$ ;
$\arccos$ : $[-1, 1]$ ;
$\arctan$ : $(-\infty, \infty)$

Sine restricted to $\left[\tfrac{\pi}{2}, \tfrac{3\pi}{2}\right]$ is
strictly ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ on that interval, which means it is one-to-one and could
serve as a valid (though non-standard) domain restriction for defining
an inverse sine function.

behavior of sine on the interval:

Procedures

What are the domain and range of $\arccos$ ?

Domain: $[-1, 1]$ ; Range: $[0, \pi]$

Domain: $[-1, 1]$ ; Range: $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$

Domain: $[0, \pi]$ ; Range: $[-1, 1]$

Domain: $(-\infty, \infty)$ ; Range: $[0, \pi]$

Which statement about the domain of $\arctan$ is correct?

The domain of $\arctan$ is all real numbers.

The domain of $\arctan$ is $[-1, 1]$ .

The domain of $\arctan$ is $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$ .

The domain of $\arctan$ is $[0, \pi]$ .

Evaluate $\arcsin\!\left(\tfrac{1}{2}\right)$ .
Express your answer as a fraction of $\pi$ (e.g., enter $\pi/6$ as
"pi/6").

Evaluate $\arccos\!\left(-\dfrac{\sqrt{3}}{2}\right)$ .
Express your answer as a fraction of $\pi$ (e.g., enter $5\pi/6$ as
"5pi/6").

Evaluate $\arctan(-1)$ .
Express your answer as a fraction of $\pi$ (e.g., enter $-\pi/4$ as
"-pi/4").

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