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Question

# sketch a graph of the function and compare the graph of g with the graph of f(x) = arcsin x. g(x) = arcsin (x - 1)

Solution

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Given funcitons

$f\left(x\right)=\arcsin x$

$g\left(x\right)=\arcsin \left(x-1\right)$

We need to graph this functions and to compare the graph of $g$ with the graph of $f$.

We know that

$\color{#c34632}{\text{The domain of f is : } -1 \leq x \leq 1}$

$\color{#c34632}{\text{The range of f is : } -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}}$

Therefore,

$\color{#4257b2}{\text{The domain of g is : } -2 \leq x \leq 0}$

$\color{#4257b2}{\text{The range of g is : } -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}}$

For $x=2$ :

$g\left( 2 \right)=\arcsin \left(2-1\right)=\arcsin 1=\frac{\pi}{2}$

For $x=0$ :

$f\left(0\right)=\arcsin 0=0$

$g\left(0\right)=\arcsin \left(0-1\right)=\arcsin \left(-1\right)=-\frac{\pi}{2}$

For $x=\pm 1$ :

$f\left(\pm 1\right)=\arcsin \left(\pm 1 \right)=\pm \frac{\pi}{2}$

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