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A local photocopying store advertises as follows. "We charge 8 \& per copy for 100 copies or less, 6 e per copy for each copy over 100 but not over $1000,$ and $4 \&$ per copy for all over $1000 . "$ Let $x$ be the number of copies ordered and $C(x)$ be the cost of the job. Write $C(x)$ algebraically and plot its graph.

$$C(x)=\left\{\begin{array}{cc}0.08 x & x \leq 100 \\2+0.06 x & 100<x \leq 1000 \\22+0.04 x & x>1000\end{array}\right.$$

Algebra

Chapter 1

Functions and their Applications

Section 3

Applications of Linear Functions

Functions

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Baylor University

University of Michigan - Ann Arbor

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Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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using a piecewise function, we're going to be determining the cost of printing. We are given here that a copying shop charges eight cents per copy to print fewer than 100 copies or 100 or less copies, six cents per copy to print over 100 copies, but not over 1000 copies and four cents per copy to print over 1000 copies. Here we can find that our limits are X must be less than or equal to 100. That X must be greater than 100 less than or equal to 1000 and the X must be greater than 1000. Knowing these limits, we can now go ahead and derive our functions, our first one being that costs eight cents her copy. And that's going to be it. That is, if X is less than or equal to 100. Our second piece, we see that it costs six cents per copy. But that is only if X is greater than 100 and less than or equal 2000. So we need to take into account the cost of printing those 1st 100 copies and that we can find just by plugging 100 into the first function we just found for X. By doing that, we find that we have an intercept equal to eight. Our third piece. We see that we have four cents per copy for any copies printed over 1000. But again, we have to take into account not only our 1st $8 for the 1st 100 but then also those 899 copies that were printed in our last limit. So by doing that, we can plug 8 99 into X here. In doing that, we end up getting a value of 53.94 Remember, we also have that 1st $8 adding those together we end up with an intercept equal to 61 94. That gives us our intercept of our third piece, 61 94. That would be the piecewise function that describes this relationship. If we wanted to graph it, it would look something like this. We would have up to 100 right there with a slope of 0.8 our next one from 100 all the way to 1000 we would have a slope of 0.6 and for any price above 1000 we would have a slope of 10000.0 forest. You can see our slope is just progressively getting flatter. As we go from our accurate graph, you can plug these functions into your graphing calculator, and that will show you exactly what it would look like.

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