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


$\begin{equation} \begin{aligned} f(x) =& -2x - 2 \text{ for } -2 \leq x \leq -1\\ f(x) =& \sqrt{1 - x^2} \text{ for } -1 \leq x \leq 1 \end{aligned} \end{equation}$