how to graph inverse trig functions

before you start, do how to evaluate inverse trig functions

a inverse trig function is in the form $f(x)=A{\sin }^{-1}(n(x-b))+c$

in a graph of a inverse trig function, the amplitude is expressed differently, because it affects the range of the graph

always remember to calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$

worked example: graph $f(x)={\sin }^{-1}(2-x)$
domain:
range: $[-\frac{\pi }{2},\frac{\pi }{2}]$
mean: $f(x)=0$

worked example: graph $f(x)=\frac{1}{2}{\sin }^{-1}(1+2x)$
to find the domain, we need to realize that $-1\le 1+2x\le 1$
$-2\le 2x\le 0$
$-1\le x\le 0$
calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$
range: $[-\frac{\pi }{4},\frac{\pi }{4}]$

worked example: graph $f(x)=2{\sin }^{-1}(2x+4)+\frac{\pi }{2}$
to find the domain, we need to realize that $-1\le 2x+4\le 1$
$-5\le 2x\le -3$
$-\frac{5}{2}\le -x\le -\frac{3}{2}$
domain: $\left[-\frac{5}{2},-\frac{3}{2}\right]$
calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$
range: $\left[-\frac{\pi }{2},\frac{3\pi }{2}\right]$

worked example: graph $f(x)={\cos }^{-1}(\frac{x}{2})-\frac{\pi }{6}$
to find the domain, we need to realize that $-1\le \frac{x}{2}\le 1$
$-2\le x\le 2$
calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$
range: $\left[-\frac{\pi }{6},\frac{5\pi }{6}\right]$

worked example: graph $f(x)=-2{\cos }^{-1}(3-2x)+\frac{\pi }{4}$
to find the domain, we need to realize that $-1\le 3-2x\le 1$
$-4\le -2x\le -3$
$-2\ge x\ge -\frac{3}{2}$
calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$
range: $\left[\frac{5\pi }{4},\frac{3\pi }{4}\right]$

worked example: graph $f(x)={\tan }^{-1}(2x)+\frac{\pi }{2}$
the domain of inverse tan is $\mathbb{R}$
calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$
range: $[0,\pi ]$
we need the x coordinate of the point of inflection, this is found by solving $2x=0$

worked example: graph $f(x)=\frac{1}{2}{\tan }^{-1}(\frac{x}{2}-1)-\frac{\pi }{4}$
the domain of inverse tan is $\mathbb{R}$
calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$
range: $[-\frac{\pi }{2},0]$
we need the x coordinate of the point of inflection, this is found by solving $\frac{x}{2}-1=0$

worked example: graph $f(x)={\tan }^{-1}(1-2x)+\pi$
the domain of inverse tan is $\mathbb{R}$
calculate range using $\left[-\frac{A\pi }{2}+c,\frac{A\pi }{2}+c\right]$
range: $[\frac{\pi }{2},\frac{3\pi }{2}]$
we need to x coordinate of the point of inflection, this is found by solving $1-2x=0$