Function Properties

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These problems are intended to build up an intuition for how the different inverse functions are evaluated. Specifically, general points such as valid domains.

Remember as you go through these problems that if $f$ is some function that has a valid inverse, which we will call $g$ , it must be the case that the range of $f$ is the same as the domain of $g$ . So, for any given $a$ we know that $g(a)$ is only going to be defined if there is some $\alpha$ with $f(\alpha )=a$ .

1 :

What is the domain of sine?

$(-\infty ,-1]\cup [1,\infty )$ Every real number that is not of the form $\frac {\pi }{2}+n\pi$ All real numbers $[-1,1]$

What is the range of sine?

$(-\infty ,-1]\cup [1,\infty )$ Every real number that is not of the form $\frac {\pi }{2}+n\pi$ All real numbers $[-1,1]$

1.1 :

As a result, what can we say about the domain of arcsine?

The domain of $\arcsin$ is $(-\infty ,-1]\cup [1,\infty )$ Every real number that is not of the form $\frac {\pi }{2}+n\pi$ The domain of $\arcsin$ is all real numbers The domain of $\arcsin$ is $[-1,1]$

1.1.1 :

So, which of the following are valid expressions?

$\arcsin (55)$ $\arcsin (\frac {\pi }{e})$ $\arcsin (\frac {e}{\pi })$ $\arcsin (0)$ $\arcsin (-7)$ $\arcsin (\frac {2}{3})$ $\arcsin (10^{-70})$ $\arcsin (\tan (\frac {5\pi }{12}))$ (Hint: You will want to consider what tangent does between $\frac {\pi }{4}$ and $\frac {\pi }{2}$ ) $\arcsin (1)$

2 :

What is the domain of secant?

$(-\infty ,-1]\cup [1,\infty )$ Every real number that is not of the form $\frac {\pi }{2}+n\pi$ All real numbers $[-1,1]$

What is the range of secant?

$(-\infty ,-1]\cup [1,\infty )$ Every real number that is not of the form $\frac {\pi }{2}+n\pi$ All real numbers $[-1,1]$

2.1 :

As a result, what can we say about the domain of arcsecant?

The domain of $\arcsec$ is $(-\infty ,-1]\cup [1,\infty )$ The domain of $\arcsec$ is every real number that is not of the form $\frac {\pi }{2}+n\pi$ The domain of $\arcsec$ is all real numbers The domain of $\arcsec$ is $[-1,1]$

2.1.1 :

So, which of the following are valid expressions?

$\arcsec (55)$ $\arcsec (\frac {\pi }{e})$ $\arcsec (\frac {e}{\pi })$ $\arcsec (0)$ $\arcsec (-7)$ $\arcsec (\frac {2}{3})$ $\arcsec (10^{-70})$ $\arcsec (\tan (\frac {5\pi }{12}))$ (Note: You will want to consider what tangent does between $\frac {\pi }{4}$ and $\frac {\pi }{2}$ ) $\arcsec (1)$

3 :

What is the domain of tangent?

$(-\infty ,-1]\cup [1,\infty )$ Every real number that is not of the form $\frac {\pi }{2}+n\pi$ All real numbers $[-1,1]$

What is the range of tangent?

$(-\infty ,-1]\cup [1,\infty )$ Every real number that is not of the form $\frac {\pi }{2}+n\pi$ All real numbers $[-1,1]$

3.1 :

As a result, what can we say about the domain of arctangent?

The domain of $\arctan$ is $(-\infty ,-1]\cup [1,\infty )$ The domain of $\arctan$ is all rational numbers The domain of $\arctan$ is all real numbers The domain of $\arctan$ is $[-1,1]$

3.1.1 :

So, which of the following are valid expressions?

$\arctan (55)$ $\arctan (\frac {\pi }{e})$ $\arctan (\frac {e}{\pi })$ $\arctan (0)$ $\arctan (-7)$ $\arctan (\frac {2}{3})$ $\arctan (10^{-70})$ $\arctan (\tan (\frac {5\pi }{12}))$ (Note: You will want to consider what tangent does between $\frac {\pi }{4}$ and $\frac {\pi }{2}$ ) $\arctan (1)$

The problem that we still need to address is how the range of the inverse functions interact with that domain. Remember that in order for the inverse functions to be functions we have to restrict what angles they will actually tell us about. So while $\arcsin (\sin (\frac {3\pi }{4}))$ is a valid expression and will tell us an angle whose sine is $\frac {1}{\sqrt {2}}$ it will not actually be $\frac {3\pi }{4}$ .

4 :

If $\theta$ is an angle whose terminal side is in quadrant 1:

In which quadrant is $\arcsin (\sin (\theta ))$ ?

1 2 3 4

If $\theta$ is an angle whose terminal side is in quadrant 2:

In which quadrant is $\arcsin (\sin (\theta ))$ ?

1 2 3 4

If $\theta$ is an angle whose terminal side is in quadrant 3:

In which quadrant is $\arcsin (\sin (\theta ))$ ?

1 2 3 4

If $\theta$ is an angle whose terminal side is in quadrant 4:

In which quadrant is $\arcsin (\sin (\theta ))$ ?

1 2 3 4

5 :

If $\theta$ is an angle whose terminal side is in quadrant 1:

In which quadrant is $\arccos (\cos (\theta ))$ ?

1 2 3 4

If $\theta$ is an angle whose terminal side is in quadrant 2:

In which quadrant is $\arccos (\cos (\theta ))$ ?

1 2 3 4

If $\theta$ is an angle whose terminal side is in quadrant 3:

In which quadrant is $\arccos (\cos (\theta ))$ ?

1 2 3 4

If $\theta$ is an angle whose terminal side is in quadrant 4:

In which quadrant is $\arccos (\cos (\theta ))$ ?

1 2 3 4