1.3 Creating New Polynomial Functions: An Introduction to ...

Example 1
Subtracting Two Polynomial Functions
Let f (x) 2(x 1)(x 2)(x 4) and g(x) (x 1)(x 2)(x 4).
(a) On the same axes, sketch the graph of each function. Use intercepts, the
degree of the function, and the leading coefficient to develop the sketch.
(b) Graph h(x) f (x) g(x).
(c) Identify the domain of each function.
9
6
3
–3 –2 –1
–3
–6
–9
–12
–15
–18
y
Solution
y = g(x)
x
1 2 3 4 5 6
y = f(x)
(a) The zeros for both f (x) and g(x) are 1, 2, and 4.
By inspection, both functions are degree-3 polynomials.
The leading coefficient of f (x) is negative.
Therefore, as x → ∞, f (x) →∞and as x →∞, f (x) → ∞.
For g(x), the leading coefficient is positive.
Therefore, as x → ∞, g(x) → ∞ and as x →∞, g(x) →∞.
The y-intercept of g(x) is 8 and the y-intercept of f (x) is 16.
Using this information results in the sketch to the left.
y
12
y = h(x)
8
y = g(x)
4
x
–3 –2 –1
–4
–8
–12
1 2 3 4 5 6
–16
–20
y = f(x)
–24
–28
–32
f (x), g(x), and h(x) f (x) g(x)
(b) h(x) f (x) g(x)
To find points that lie on h(x), subtract the y-coordinates of points
that have the same x-coordinate. The difference function has the
same zeros as the two original functions, that is, 1, 2, and 4.
Subtracting the y-intercepts results in point (0, 24). Create a table
to record some additional key points on f and g, and use these to
find more points on h.
x f (x) g(x) h(x) f (x) g(x)
1 12 6 12 6 18
3 8 4 8 (4) 12
5 36 18 36 18 54
(c) Since f (x) and g(x) are polynomial functions, their domains are the same,
that is, the set of real numbers. The difference function h(x) is also a
polynomial function, and its domain is also the set of real numbers.
For any two functions, the y-values can only be added or subtracted when
the x-values are common. For any polynomial function, the domain is the set
of real numbers, so the difference, or sum, function will have the same
domain, the set of real numbers. However, the range of f (x) ± g(x) must be
determined each time.
1.3 CREATING NEW POLYNOMIAL FUNCTIONS: AN INTRODUCTION TO COMPOSITION 29