Question

Write a piecewise function for the given graph.

Answer 1

A piecewise function that represents the graph above is
`f(x)=\left\{\begin{array}{Ir}x - 2, & \text{if } x \geq -1 \\-x, & \text{if } x < -1\end{array}\right`.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

`y - y_1 = m(x - x_1)`

Where:

• x and y represent the data points.
• m represent the slope.

First of all, we would determine the slope of the left line;

`Slope(m)=(y_2-y_1)/(x_2-x_1)`

Slope (m) = (1 - 8)/(-1 + 8)

Slope (m) = -7/7 = -1

At data point (-1, 1) and a slope of -1, an equation for this line can be calculated by using the point-slope form as follows:

y - 1 = -1(x + 1)

y = -x, if x < -1 (since the dot is hollow at -1).

For line 2, we have;

Slope (m) = (2 + 3)/(4 + 1)

Slope (m) = 5/5 = 1

At data point (0, -2) and a slope of 1, an equation for this line can be calculated by using the point-slope form as follows:

y + 2 = 1(x - 0)

y = x - 2, if x ≥ -1 (since the dot is solid at -1).

Therefore, the piecewise function can be written as follows;

`f(x)=\left\{\begin{array}{Ir}x - 2, & \text{if } x \geq -1 \\-x, & \text{if } x < -1\end{array}\right`