2.1
INPUT AND OUTPUT
1
Finding Output Values:
Evaluating a Function
Example 4
Let h(x) = x2 − 3x + 5. Evaluate and simplify the following
expressions.
(a) h(2)
(b) h(-3)
(c) h(a)
(d) h(a) − h(2)
Solution
Notice that x is the input and h(x) is the output. It is helpful
to rewrite the formula as
Output = h(Input) = (Input)2 − 3・ (Input) + 5.
(a) For h(2), we have Input = 2, so
h(2) = (2)2 − 3・(2) + 5 = 3.
2
Finding Output Values:
Evaluating a Function
Example 4 continued
h(x) = x2 − 3x + 5.
Evaluate (b) h(−3) (c) h(a)
(d) h(a) − h(2)
Solution
(b) In this case, Input = −3. We substitute and simplify
h(−3) = (−3)2 − 3(−3) + 5 = 9 + 9 + 5 = 23.
(c) In this case, Input = a: h(a) = (a)2 − 3(a) + 5 = a2 − 3a + 5 .
(d) Using h(2) = 3 from part (a):
h(a) − h(2) = a2 − 3a + 5 − 3 = a2 − 3a + 2.
3
Finding Input Values:
Solving Equations
Example 6 (a)
Suppose
1
f ( x) 
x4
(a) Find an x-value that results in f(x) = 2.
Solution (a)
To find an x-value that results in f(x) = 2, solve the equation
1
x4
1
4 
x4
4( x  4)  1
2 
Squaring both sides
Multiplying both sides by x - 4
4 x  16  1
x  17 / 4  4.25
4
Finding Output Values
from Tables
Example 8(a)
The table shows the revenue, R = f(t), received, by the National
Football League, NFL, from network TV as a function of the year,
t, since 1975. (a) Evaluate and interpret f(25).
Year, t (since 1975)
Revenue, R (million $)
0
5
10
15
20
25
30
201
364
651
1075
1159
2200
2200
Solution (a)
The table shows f(25) = 2200. Since t = 25 in the year 2000, we
know that NFL’s revenue from TV was $2200 million in the year
2000.
5
Finding Input/Output Values
from Graphs
Exercise 35 (a) and (c)
An epidemic of influenza spreads through
a city. The figure to the right shows the
graph of I = f(w), where I is the number of
individuals (in thousands) infected w
weeks after the epidemic begins.
9
8
7
6
5
4
3
2
1
0
f
f(w)
0
2
4
6
8
10
12
14
16
w
(a) Evaluate f(2) and explain its meaning .
(b) Solve f(w) = 4.5 and explain what the solutions means.
Solution
(a) f(2) ≈ 7, so 7 thousand individuals were infected 2 weeks after onset.
(b) If you imagine a horizontal line drawn at f = 4.5, it would intersect
the graph at two points, at approximately w = 1 and w = 10. This would
mean that 4.5 thousand individuals were infected with influenza both
6
one week and ten weeks after the epidemic began.