### Home > APCALC > Chapter 4 > Lesson 4.2.5 > Problem4-103

4-103.

For $f\left(x\right)=\sin\left(x^2\right)$, $g ( x ) = \sqrt { x - 2 }$, and $h(x)=\frac{1}{x}$, write equations for each of the following compositions of functions and state the domains.

1. $f\left(g\left(x\right)\right)$

Since $g$ is a square root function, the domain of $g$ is $x ≥ 2.$

$f(g(x))=\sin(\sqrt{x-2}^2)=$

2. $f\left(g\left(h\left(x\right)\right)\right)$

$h$ is a rational function, so the denominator cannot equal $0$.

Before writing your simplified equation, the quantity inside of the square root must be greater than or equal to $0$.

3. $h\left(f\left(g\left(x\right)\right)\right)$

$h(\text{part (a)})=\frac{1}{\text{(part (a))}}=$

The denominator cannot equal $0$. For what value(s) of $x$ does $\sin\left(x−2\right)=0$? Also, due to the square root in the original function, $x – 2 ≥ 0$.

Domain: $x>2$ and $x ≠ 2+πn$, where $n$ is an integer value

4. $h\left(h\left(x\right)\right)$

$h(h(x))=\frac{1}{\frac{1}{x}}$