Page 176 - ArithBook5thEd ~ BCC

Example 221. Let f be the linear function defined by

f (x)= −2x +300.

Find f (0), f (10), f (25) and f (150).

Solution. The function takes an input x,multiplies it by −2, and then adds 300. Hence

f (0) = −2(0) + 300 = 300
f (10) = −2(10) + 300 = 280

f (25) = −2(25) + 300 = 250
f (150) = −2(150) + 300 = 0.

Example 222. New York City Taxi fares are calculated as follows: there is an initial charge of $2.50.
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After that, there is a charge of $0.50 per mile (and $0.50 per minute in slow traffic). Assuming
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there is no traffic (!), write a linear function which expresses how the taxi fare depends on the distance
travelled.

Solution. Let x denote the distance travelled (in miles), and let F(x) be the function that determines
the corresponding fare. You always pay the initial charge, so F(x)is at least $2.50, even if you go
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nowhere. Then you are charged $0.50 for every mile. So for one mile you pay 5 × $0.50 = $2.50. If
5
you go x miles, you pay x times $2.50, or 2.50x dollars. Putting this all together,

F(x)= 2.50 + 2.50x.

There are many other types of function. A quadratic function has the form

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q(x)= ax + bx + c,

where a, b and c are three fixed numbers, and a is not 0 (otherwise, it’s a linear function.) For example,
the quadratic function
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h(t)= −16t +100
expresses the height (h) above the ground of an object dropped from a 100-foot tower after t seconds.
Here, the height h is a function of the time t.
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Example 223. Using the height function h(t)above, find the height of an object 1 seconds after it
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is dropped from a 100-foot tower.

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