Find the domain and range for the indicated function.
-If $f ( x ) = \sqrt { x } , g ( x ) = \frac { x } { 3 }$, and $h ( x ) = 3 x + 9$, find $f ( g ( h ( x ) ) )$
A) $\sqrt { x + 9 }$
B) $\sqrt { x + 3 }$
C) $\sqrt { x } + 3$
D) $3 \sqrt { x } + 9$

Question

Quizplus · Accepted Answer

First we need to find the composition of the three functions inside the parentheses of $f(g(h(x)))$. Starting from the inside and working outwards, we have:

$$h(x) = 3x + 9$$
$$g(h(x)) = \frac{3x+9}{3} = x+3$$
$$f(g(h(x))) = \sqrt{x+3}$$

Thus, the answer is $\boxed{	extbf{(B)}\ \sqrt { x + 3 }}$.

To find the domain and range of $f(x) = \sqrt{x}$, note that the radicand ($x$) must be nonnegative for the function to be defined. Therefore, the domain is $[0,\infty)$. The range consists of all nonnegative real numbers, so the range is also $[0,\infty)$.

To find the domain and range of $g(x) = \frac{x}{3}$, note that the denominator is nonzero for all $x$, so the only restriction on the domain is that $x$ must be a real number (i.e. there is no domain restriction). The range consists of all real numbers, so the range is $(-\infty,\infty)$.

To find the domain and range of $h(x) = 3x+9$, note that there is no domain restriction (i.e. all real numbers are in the domain). The range consists of all real numbers, so the range is $(-\infty,\infty)$.

Find the Domain and Range for the Indicated Function f(x)=x,g(x)=x3f ( x ) = \sqrt { x } , g ( x ) = \frac { x } { 3 }f(x)=x​,g(x)=3x​

Find the Domain and Range for the Indicated Function $f ( x ) = \sqrt { x } , g ( x ) = \frac { x } { 3 }$