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For $f\left(x\right)={x}^{2}+3$ and $g\left(x\right)=2x-5,$ find $\left(f\text{/}g\right)\left(x\right)$ and state its domain.
$\left(\frac{f}{g}\right)\left(x\right)=\frac{{x}^{2}+3}{2x-5}.$ The domain is $\left\{x|x\ne \frac{5}{2}\right\}.$
When we compose functions, we take a function of a function. For example, suppose the temperature $T$ on a given day is described as a function of time $t$ (measured in hours after midnight) as in [link] . Suppose the cost $C,$ to heat or cool a building for 1 hour, can be described as a function of the temperature $T.$ Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating $C\left(T\left(t\right)\right).$ We have defined a new function, denoted $C\circ T,$ which is defined such that $\left(C\circ T\right)\left(t\right)=C(T\left(t\right))$ for all $t$ in the domain of $T.$ This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function $(C\circ T)(t).$ It does not make sense to consider $(T\circ C)(t),$ because temperature is not a function of cost.
Consider the function $f$ with domain $A$ and range $B,$ and the function $g$ with domain $D$ and range $E.$ If $B$ is a subset of $D,$ then the composite function $(g\circ f)(x)$ is the function with domain $A$ such that
A composite function $g\circ f$ can be viewed in two steps. First, the function $f$ maps each input $x$ in the domain of $f$ to its output $f(x)$ in the range of $f.$ Second, since the range of $f$ is a subset of the domain of $g,$ the output $f(x)$ is an element in the domain of $g,$ and therefore it is mapped to an output $g\left(f\left(x\right)\right)$ in the range of $g.$ In [link] , we see a visual image of a composite function.
Consider the functions $f(x)={x}^{2}+1$ and $g(x)=1\text{/}x.$
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