College Algebra, Chapter 3, 3.6, Section 3.6, Problem 64

Levi have a $\$50$ coupon from the manufaturer good for the purchase of a cellphone. The store where Levi is purchasing his cellphone is offering a 20 percent discount on all cellphones. Let $x$ represents the regular price of the cellphone.
a.) If only the $20 \%$ discount applies, then find a function $f$ that models the purchase price of the cellphone as a function of the regular price $x$.

$
\begin{equation}
\begin{aligned}
f(x) &= x - 0.2 x && \text{Where } x \text{ is the regular price}\\
\\
f(x) &= 0.8x
\end{aligned}
\end{equation}
$


b.) If only the $\$50$ coupon applies, then find a function of $g$ that models the purchase price of the cellphone as a function of the sticker price $x$.
$g(x) = x - 50$ where $x$ is the regular price and $50$ is the discount coupon.

c.) If you can use the coupon and the discount, then the purchase price is either $f \circ g(x)$ or $g \circ f(x)$, depending on the order in which they are applied to the price. Find both $f \circ g(x)$ and $g \circ f(x)$. Which composition gives the lower price?
For $f \circ g(x)$,

$
\begin{equation}
\begin{aligned}
f \circ g(x) &= f(g(x)) && \text{Definition of } f \circ g\\
\\
f \circ g(x) &= f (x - 50) && \text{Definition of } g\\
\\
f \circ g(x) &= 0.8(x-50) && \text{Definition of } f
\end{aligned}
\end{equation}
$


For $g \circ f(x)$,

$
\begin{equation}
\begin{aligned}
g \circ f(x) &= g(f(x)) && \text{Definition of } g \circ f\\
\\
g \circ f(x) &= g(0.8x) && \text{Definition of } f\\
\\
g \circ f(x) &= 0.8x-50 && \text{Definition of } g
\end{aligned}
\end{equation}
$

$g \circ f(x)$ gives the lower price.