(2.5) Domain and Range of Composite Functions
Consider a function $h\left(x\right)=x$.
a) State the domain of $h\left(x\right)$. $\{x|$
$\}$
b) State the range of $h\left(x\right)$. $\{y|$
$\}$
Consider a composite function where $f(x)=x^2$ and $g(x)=\sqrt{x}$.
$g\left(f\left(x\right)\right)=$
$g\left(f\left(x\right)\right)=x$ but is the domain $x\in R$ and the range $y\in R$?
a) Look at the diagram. State the domain of $f\left(x\right)$. $\{x|$
$\}$
b) Look at the diagram. State the range of $f\left(x\right)$. $\{y|$
$\}$
c) If these become the inputs for $g\left(x\right)$, what is the range of $\left(g\circ f\left(x\right)\right)$? $\{y|$
$\}$
In summary, even though $g\left(f\left(x\right)\right)=x$, the domain of $g\left(f\left(x\right)\right)$ is $\{x|$
$\}$
and the range of $g\left(f\left(x\right)\right)$ is $\{y|$
$\}$
Consider a composite function where $f\left(x\right)=x^2$ and $g\left(x\right)=\sqrt{x}$.
$f\left(g\left(x\right)\right)=$
$f\left(g\left(x\right)\right)=x$ but is the domain $x\in R$ and the range $y\in R$?
a) Look at the diagram. State the domain of $g\left(x\right)$. $\{x|$
$\}$
b) Look at the diagram. State the range of $g\left(x\right)$. $\{y|$
$\}$
c) If these become the inputs for $f\left(x\right)$, what is the range of $\left(f\circ g\left(x\right)\right)$? $\{y|$
$\}$
In summary, even though $f\left(g\left(x\right)\right)=x$, the domain of $f\left(g\left(x\right)\right)$ is $\{x|$
$\}$
and the range of $f\left(g\left(x\right)\right)$ is $\{y|$
$\}$