Linear function (calculus): Difference between revisions

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Revision as of 10:54, 30 January 2021

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===Example===

Suppose salami and sausage cost €6 and €3 per kilogram, and we wish to buy €12 worth. How much of each can we purchase? If ''x'' kilograms of salami and ''y'' kilograms of sausage costs a total of €12 then, €6*€6×''x'' + €3*€3×''y'' = €12. Solving for ''y'' gives the point-slope form <math>y = -2x + 4</math>, as above. That is, if we first choose the amount of salami ''x'', the amount of sausage can be computed as a function <math>y = f(x) = -2x + 4</math>. Since salami costs twice as much as sausage, adding one kilo of salami decreases the sausage by 2 kilos: <math>f(x{+}1) = f(x) - 2</math>, and the slope is &minus;2. The ''y''-intercept point <math>(x,y)=(0,4)</math> corresponds to buying only 4kg of sausage; while the ''x''-intercept point <math>(x,y)=(2,0)</math> corresponds to buying only 2kg of salami.

Note that the graph includes points with negative values of ''x'' or ''y'', which have no meaning in terms of the original variables (unless we imagine selling meat to the butcher). Thus we should restrict our function <math>f(x)</math> to the ___domain <math>0\le x\le 2</math>.