Single Variable Calculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 18

Express the function consisting of a line segment from the point $(-2,2)$ to the point $(-1,0)$ together with the top half of the circle with center at the origin and radius 1.

We can get the expression of a line segment using the two point form.


$
\begin{equation}
\begin{aligned}

y - y_1 =& \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)\\
y - 2 =& \frac{0 - 2}{-1-(-2)} (x - (-2))\\
y - 2 =& \frac{-2}{1}(x + 2)\\
y = & -2x - 2 \text{ for } -2 \leq x \leq -1

\end{aligned}
\end{equation}
$



The equation for the top half of circle with radius 1 and center at origin is...


$
\begin{equation}
\begin{aligned}

x^2 + y^2 =& 1^2\\
y^2 =& 1 -x^2\\
y =& \sqrt{1 - x^2} \text{ for } -1 \leq x \leq 1

\end{aligned}
\end{equation}
$


Therefore, the expression for this function is


$
\begin{equation}
\begin{aligned}

f(x) =& -2x - 2 \text{ for } -2 \leq x \leq -1\\
f(x) =& \sqrt{1 - x^2} \text{ for } -1 \leq x \leq 1

\end{aligned}
\end{equation}
$