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EVALUATE
POLYNOMIAL FUNCTIONS
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Objectives
2
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Definition of a Polynomial Function
Polynomial Function
A polynomial function of degree n is defined by
f (x) = an xn + an – 1 xn – 1 + · · · + a1 x + a0 ,
for real numbers an,an – 1, . . . , a1, and a0 , where an ≠ 0 and n is a whole
number.
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EXAMPLE 1
Evaluating Polynomial Functions
Let f(x) = 4x3 – 5x2 + 7. Find each value.
(a) f(2)
f(x) = 4x3 – 5x2 + 7
f(2) = 4 • 23 – 5 • 22 + 7
= 4 • 8 – 5 • 4 + 7
= 32 – 20 + 7
= 19
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EXAMPLE 1
Evaluating Polynomial Functions
Let f(x) = 4x3 – 5x2 + 7. Find each value.
(b) f(–3)
f(x) = 4x3 – 5x2 + 7
f(–3) = 4 • (–3)3 – 5 • (–3)2 + 7
= 4 • (–27) – 5 • 9 + 7
= –108 – 45 + 7
= –146
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6
Functions
While f is the most common letter used to represent functions, recall that
other letters such as g and h are also used. The capital letter P is often used
for polynomial functions.
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EXAMPLE 2
Using a Polynomial Model to Approximate Data
The number of U.S. households estimated to see and pay at least one bill
on-line each month during the years 2000 through 2006 can be modeled by
the polynomial function defined by
P(x) = 0.808x2 + 2.625x + 0.502,
where x = 0 corresponds to the year 2000, x = 1 corresponds to 2001, and
so on, and P(x) is in millions. Use this function to approximate the number
of households expected to pay at least one bill on-line each month in 2006.
Since x = 6 corresponds to 2006, we must find P(6).
P(x) = 0.808x2 + 2.625x + 0.502
P(6) = 0.808(6)2 + 2.625(6) + 0.502
= 45.34
Thus, in 2006 about 45.34 million households are expected to pay at least
one bill on-line each month.
Let x = 6.
Evaluate.
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Adding and Subtracting Functions
Adding and Subtracting Functions
If f(x) and g(x) define functions, then�
(f + g) (x) = f (x) + g(x) Sum function
and (f – g) (x) = f (x) – g(x). Difference function
In each case, the domain of the new function is the intersection of the
domains of f(x) and g(x).
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EXAMPLE 3
Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 2x2 – 3x + 4 and g(x) = x2 + 9x – 5,
find (a) the sum and (b) the difference.
(a) (f + g) (x) = f (x) + g(x) Use the definition.
= (2x2 – 3x + 4) + (x2 + 9x – 5) Substitute.
= 3x2 + 6x – 1 Add the polynomials.
(b) (f – g) (x) = f (x) – g(x) Use the definition.
= (2x2 – 3x + 4) – (x2 + 9x – 5) Substitute.
= (2x2 – 3x + 4) + (–x2 – 9x + 5) Change subtraction
to addition.
= x2 – 12x + 9 Add.
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EXAMPLE 4
Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(a) (f + g) (5)
(f + g) (5) = f (5) + g(5) Use the definition.
= [4(5)2 – 5] + 3(5) Substitute.
= 110
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11
Sec 6.3 - 11
EXAMPLE 4
Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(b) (f – g) (x) and (f – g) (3)
(f – g) (x) = f (x) – g(x) Use the definition.
= (4x2 – x) – 3x Substitute.
= 4x2 – 4x Combine like terms.
Then,
(f – g) (3) = 4(3)2 – 4(3) = 24. Substitute.
Confirm that f (3) – g(3) gives the same result.
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The simplest polynomial function is the Linear function, defined by f(x) = x.
x
y
x
f(x) = x
–2
–1
0
1
2
–2
–1
0
1
2
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The Quadratic function, is defined by f(x) = x2.
x
y
x
f(x) = x2
–2
–1
0
1
2
4
1
0
1
4
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The cubic function, is defined by f(x) = x3.
x
y
x
f(x) = x3
–2
–1
0
1
2
–8
–1
0
1
8
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EXAMPLE 5
x
f(x) = –2x
Graphing Linear Function
Graph the function by creating a table of ordered pairs. Give the domain and the range of
the function by observing the graph.
(a) f(x) = –2x.
x
y
–2
–1
0
1
2
4
2
0
–2
–4
Domain
Range
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EXAMPLE 5
x
f(x) = x2 – 2
Graphing Quadratic Function
Graph the function by creating a table of ordered pairs. Give the domain and
the range of the function by observing the graph.
(b) f(x) = x2 – 2.
x
y
–2
–1
0
1
2
2
–1
–2
–1
2
Domain
Range
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