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# In Exercise given below, use f(x) and g(x) to find each composition. Identify its domain. Do not use a calculator. (a) $(f \circ g)(x) \quad$ (b) $(g \circ f)(x) \quad$ (c) $(f \circ f)(x)$$f(x)=\frac{x-3}{2}, g(x)=2 x+3$

Solution

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Using the given value of $f(x)$ and $g(x)$, we will determine the value of the given functions.

$\textbf{Part (a),}$

Knowing that:

$(f\cdot g)(x)=f(g(x))$

By definition:

\begin{aligned} (f\cdot g)(x)&=f(2x+3)\\ &=\frac{(2x+3)-3}{2}\\ &=\frac{2x+3-3}{2}\\ &=\frac{2x}{2}\\ &\boxed{=x} \end{aligned}

Function $(f\cdot g)(x)=x$ is defined for all real numbers. Therefore, its domain is $(-\infin,\infin)$.

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