Chapter 1.1, Problem 52E

To determine

To find: An expression for the given graph.

Answer to Problem 52E

The expression for the function of the given graph is $\underset{\_}{f\left(x\right)=\{\begin{array}{ll}-\frac{3}{2}x-3\hfill & \text{if }-4\le x\le -2\hfill \\ \sqrt{4-x^2}\hfill & \text{if }-2<x<2\hfill \\ \frac{3}{2}x-3\hfill & \text{if 2}\le x\le 4\hfill \end{array}}$.

Explanation of Solution

Formula used:

Point slope form, $y-y_1=m\left(x-x_1\right)$.

The equation of the circle with center $\left(0,0\right)$ and radius r is, $x^2+y^2=r^2$.

Calculation:

The graph has an upper half of a circle and two lines.

Obtain the expression for an upper half of a circle with center $\left(0,0\right)$ and radius 2 on the domain $-2<x<2$.

The equation of the circle with center $\left(0,0\right)$ and radius 2 is $x^2+y^2=2^2$.

Find the value of y to obtain the equation of the upper half of the circle.

$\begin{array}{l}{x}^2+y^2=2^2\\ y^2=2^2-x^2\end{array}$

$y=\sqrt{4-x^2}$ (1)

Obtain the expression for a line with slope $-\frac{3}{2}$ passing through $\left(-2,0\right)$ on the domain $-4\le x\le -2$.

Use the point slope form and obtain the equation of the line with slope $-\frac{3}{2}$ and the point $\left(-2,0\right)$.

$y-0=-\frac{3}{2}\left(x-\left(-2\right)\right)$

$y=-\frac{3}{2}\left(x+2\right)$

$y=-\frac{3}{2}x-3$ (2)

Obtain the expression for the other line with slope $\frac{3}{2}$ and passing through $\left(2,0\right)$ on the domain $2\le x\le 4$.

Use point slope form and obtain the equation of the line with slope $\frac{3}{2}$ and the point $\left(2,0\right)$.

$\begin{array}{l}y-0=\frac{3}{2}\left(x-2\right)\\ y=\frac{3}{2}x-\frac{3}{2}\cdot 2\end{array}$

$y=\frac{3}{2}x-3$ (3)

Combine the equations (1), (2), and (3) and the domains to obtain the required function.

The piecewise function of the given graph is, $f\left(x\right)=\{\begin{array}{l}-\frac{3}{2}x-3\text{ if }-4\le x\le -2\\ \sqrt{4-x^2}\text{if }-2<x<2\\ \frac{3}{2}x-3\text{if 2}\le x\le 4\end{array}$.