## Task

Katy is told that the cost of producing $x$ DVDs is given by $C(x) = 1.25x + 2500.$ She is then asked to find an equation for $\frac{C(x)}{x}$, the average cost per DVD of producing $x$ DVDs.

She begins her work: $$\frac{C(x)}{x} =\frac {1.25x+2500}{x}$$ and finishes by simplifying both sides to get: $$C = 1.25+\frac{2500}{x}$$ Is Katy's answer correct? Explain.

## Solution

Katy has made a common error. She has interpreted the function notation, $C(x)$, as multiplication notation. She thinks: $$\frac{C(x)}{x} = \frac{C \cdot x}{x} = C$$
In this problem, $C$ is a function that uses the equation $C(x) =1.25x + 2500$ to assign to each number $x>0$ another number called, $C(x)$. $C(x)$ is the notation for the number that $C$ assigns to $x$, not the result of multiplying $C$ and $x$.

As stated, Katy could have correctly answered this question with
$$
\frac{C(x)}{x} = 1.25+ \frac{2500}{x}.
$$