Exercise 1.3.65

Question

Let $f$ and $g$ be linear functions with equations $f(x)=m_{1} x+b_{1}$ and $g(x)=m_{2} x+b_{2}$. Is $f \circ g$ also a linear function? If so, what is the slope of its graph?

Khagan Gulusoy · Accepted Answer

\begin{proof}
    To demonstrate that a function $T$ is linear, we must prove that it is representable in the form
$$
\forall x \in X: T(x)=m x+b,
$$
where $X$ is the domain of the function $T$ and $m, b \in R$ are some constants. For given functions $f$ and $g$, we have
$$
\begin{aligned}
 &f \circ g(x)=f(g(x))=f\left(m_{2} x+b_{2}ight)=m_{1}\left(m_{2} x+b_{2}ight)+b_{1}= \
& =m_{1} m_{2} x+m_{1} b_{2}+b_{1} .
\end{aligned}
$$
Letting $m:=m_{1} m_{2}$ and $b:=m_{1} b_{2}+b_{1}$, we see that $f\circ g$ is a linear function.
\end{proof}

Exercise 1.3.65

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Exercise 1.3.65

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