Question

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

• A. $[1,+\infty )$
• B. $[0,+\infty )$
• C. $[-\infty ,+\infty )$
• D. $[-1,+\infty )$

Answer 1

The correct option is B $[0,+\infty )$

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

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

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

$f\left(\sqrt{2x}\right)={\left(\sqrt{2x}\right)}^2+1=2x+1$

Thus, $\left(fog\right)\left(x\right)=f\left(g\right(x\left)\right)=2x+1$

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 square root of a negative number etc..)

In this function u can plug in all values of $x$ that is if $2x+1=0$.

That is $x=-\frac{1}{2}$

Therefore, the domain of the function is $[0,+\infty )$.