Answer
.
Work Step by Step
(a)
Hence,
$\begin{align}
& m=\frac{47.8-38.9}{30-20} \\
& =0.89
\end{align}$
It is given that the line passes through the points:
$\begin{align}
& y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) \\
& y-{{y}_{2}}=m\left( x-{{x}_{2}} \right)
\end{align}$
Then,
$\begin{align}
& y-38.9=0.89\left( x-20 \right) \\
& y-47.8=0.89\left( x-30 \right)
\end{align}$
Therefore, the equation of the line is $y-38.9=0.89\left( x-20 \right)$ or $y-47.8=0.89\left( x-30 \right)$.
(b)
Consider the equations,
$\begin{align}
& y-38.9=0.89\left( x-20 \right) \\
& y=0.89x-17.8+38.9 \\
& y=0.89x+21.1
\end{align}$
Where, $y$ is also a function of $x$. It means we can write $y=f\left( x \right)$.
Therefore, the equation of the line by use of function notation is $f\left( x \right)=0.89x+21.1$.
(c)
Consider the linear function,
$f\left( x \right)=0.89x+21.1$
Where, $x$ is the number of years after $1980$ and $f\left( x \right)$ is the percentage of never married American females.
In $2020$ ,
$\begin{align}
& x=2020-1980 \\
& =40
\end{align}$
Substitute the value of $x$ in the given equation,
$\begin{align}
& f\left( x \right)=0.89\cdot 40+21.1 \\
& =56.7
\end{align}$
Therefore, the percentage of never married American females in $2020$ is $56.7\%$.
