Question

Find the composition function $(f\circ g)\left(x\right)$ and its domain.
$f\left(x\right)=\sqrt{(}-x+1),g(x)=x^2-8$

• A. $[-2,2]$
• B. $[-1,6]$
• C. $[-3,3]$
• D. $[-\infty ,\infty ]$

Answer 1

The correct option is C $[-3,3]$

Since $f\left(x\right)=\sqrt{-x+1}$ and $g\left(x\right)=x^2-8$ and also $\left(fog\right)\left(x\right)=f\left(g\right(x\left)\right)$.

$\left(fog\right)\left(x\right)=f\left(g\right(x\left)\right)$

$\left(fog\right)\left(x\right)=f\left(\sqrt{x^2-8}\right)$ and therefore,

$f(x^2-8)=\sqrt{-(x^2-8)+1}=\sqrt{-x^2+8+1}=\sqrt{9-x^2}$

Thus, $\left(fog\right)\left(x\right)=f\left(g\right(x\left)\right)=\sqrt{9-x^2}$

Domain of a function are the values which you can plug into a function, so that you get a viable answer (the answer isn't infinity or a sqrt of a negative number etc..)

in this function you can plug in all values of $x$ that is if $\sqrt{9-x^2}=0$

That is $x^2=9$ or,

$x=3$ and $x=-3$

Therefore, the domain of the function is $[-3,3]$