Question

The graph shows the amount customers pay for x pounds of cheese at a deli counter. Customers pay a certain price per pound up to exactly 5 pounds. If a customer buys more than 5 pounds. then the price per pound for the entire purchase is discounted.

Answer 1

The amount the customers pay is an illustration of a piecewise functions

The equations of the piecewise functions are: $y = \left[\begin{array}{cc}5x&0 < x \le 5\\4x&x > 5\end{array}\right$

How to determine the equation of the piecewise function?

For the first function, we have the following points.

(x,y) = (0,0) and (4,20) for 0 < x <= 5

Since the line passes through the origin, the equation of the line would be:

$y = \frac{y_1}{x_1} * x$

So, we have:

$y = \frac{20}{4} * x$

$y = 5x$   0 < x <= 5

For the second function, we have the following points

(x,y) = (8,32) and (9,36) for x > 5

The equation of the line would be:

$y = \frac{y_1-y_2}{x_1-x_2} * (x - x_1) + y_1$

So, we have:

$y = \frac{36-32}{9-8} * (x - 8) + 32$

$y = 4(x - 8) + 32$

$y = 4x - 32 + 32$

$y = 4x$    x > 5

Hence, the equations of the piecewise functions are:

$y = \left[\begin{array}{cc}5x&0 < x \le 5\\4x&x > 5\end{array}\right$

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