6.3B – Combining Functions (Multiplying and Dividing)
Consider what happens when two functions are multiplied.
let f(x) = x + 2 and g(x) = -x2 + 5, x∈R
Example 1:
a) Sketch the two functions on the grid below.
10
8
6
4
2
0
-5
-4
-3
-2
-1 -2 0
1
2
3
4
5
-4
-6
-8
x
f(x) = x + 2
g(x) = -x2 + 5
H(x) = f(x) · g(x)
-3
-2
-1
0
1
2
3
-1
-4
4
0
1
0
1
4
4
2
5
10
3
4
12
4
1
4
5
-4
-20
-10
b) Complete the table of values above for the two functions.
c) Sketch the product of the two functions h(x) on the above grid.
d) Algebraically determine the product of the two functions. What is the degree
H(x)
= (x + 2) (-x2 + 5)
= -x3 + 5x - 2x2 + 10
= -x3 -2x2 + 5x + 10
Cubic. It seems that the new function has an overall degree equal to adding
the degree from the two functions it is a product of.
e) Use your graphing calculator to graph all functions simultaneously.
f) How does the domain of the original graphs compare to the product of the two.
Domain of product function is the same as the two original functions.
Example 2:
Determine an algebraic expression for f(x)·g(x), state degree and describe domain
a) f(x) = -2x + 3, g(x) = 3x + 4
b) f(x) = -x2 + 2, g(x) = 2x2 – 1
h(x) = f(x)·g(x)
= (-2x + 3) (3x + 4)
= -6x2 – 8x + 9x + 12
= -6x2 + x + 12
h(x) = f(x)·g(x)
= (-x2 + 2)(2x2 – 1)
= -2x4 + x2 + 4x2 – 2
= -2x4 + 5x2 – 2
Quadratic, domain x ∈ R
6.3B – combining functions (multiplying and dividing)
Quartic, domain x ∈ R
Example 3:
An economist for a Sports drink company estimates the revenue and cost
functions for the manufacture of a new drink. These functions are
R(x) = x2 + 5x and C(x) = 2x + 3, respectively, where x is the number in
thousands of drinks produced. Revenue, cost and profit are in cents.
a) Determine the average profit function, which the a quotient of two functions
r ( x) − c( x)
x
2
( x + 5 x) − (2 x + 3)
=
x
2
x + 3x − 3
=
x
AP( x) =
b) What is the average profit for 5000 drinks produced?
AP(5) = 7.4
c) What is the result when two polynomial functions are divided?
Rational functions are created (so need to be careful about restrictions?)
Example 4:
Let f(x) = x + 3 and g(x) = -x2 + 4, x∈R
a) Sketch each graph on the same axes below.
-5
Asymptote ?
4
3
2
1
0
-1 0
-2
-3
-4
x
-3
-2
5
-1
0
1
2
3
f(x)= x + 3
0
g(x)= -x2 + 4
-5
1
2
3
4
5
6
0
3
4
3
0
-5
H(x) = f(x) / g(x)
0
Undefined
(asymptote ?)
0.67
0.75
1.33
Undefined
-1.2
b) Complete the table above for -3 ≤ x ≤ 3. Use the table to sketch h(x) on the same
axes. Describe the shape of the graph.
x+3
− ( x + 3)
=
c) Determine the algebraic model for h(x). h( x) =
2
(
x
− 2)( x + 2)
−x +4
In factored
form can see
asymptotes
and intercepts
d) What is the domain of h(x)? How does the domain compare with the domains of f(x)
and g(x)? What additional consideration is there in the domain of the quotient of two
functions?
6.3B – combining functions (multiplying and dividing)
Overall interval for domain is the same but now we have certain restrictions
6.3B – Combining Function (Multiplying and Dividing) Practice Questions
1. Use the graph of f(x) and g(x) below to sketch the graph of h(x)=f(x)•g(x) and describe
max/min as well as increasing and decreasing intervals.
f(x) = x + 3
g(x) = x2 + 4
2. Use the graph of f(x) and g(x) above sketch the graph of h(x)=f(x)/g(x) and m(x)=g(x)/f(x)
then describe max/min as well as increasing and decreasing intervals.
3. Let f(x) = 3x – 1, 0 < x < 6 and g(x) = x2 – 6x, -1 < x < 4. Graph f and g on the same axes and
then use appropriate ordered pairs to sketch the graph of f · g. What is the domain of f · g?
4. Let f(x) = 2x – 3 and g(x) = 2x3 + 4x2 + 5x + 1, x∈R. Determine an expression for f(x)•g(x)
5. The speed an object travels is given by the function S(t) = 10t + 5, where speed in measured
in metre/second and the time it travels is given by the function T(t) = 2t where time is
measured in seconds. Find a function to represent the distance D(t) and then determine the
distance travelled in 10 seconds?
6. A housing development begins in 2000 as a 100-m by 50-m parcel of land. Each year its
length grows by 20 m and its width grows by 10 m.
a)
b)
c)
d)
Express the length of the development as a function of time.
Express the width as a function of time.
What function represents the area of the development in terms of time?
If the development continues to grow at the same rate what will be its area in 2010?
7. Let f(x) = mx2 + 2x + 5 and g(x) = 2x2 – nx – 2. The functions are combined to form a new
function h(x) = f(x) • g(x). Points (1,-40) and (-1,24) satisfy the new function. Determine
f(x) and g(x)
6.3B – combining functions (multiplying and dividing)
8. Find an equation in the form y =
k
so that its graph closely matches the one
x + bx + c
2
shown. Check your answer with a graphing calculator.
9. If f ( x ) =
x 3 + 27
x3 − 4x
x 2 + x − 12
3x − 1
g
(
x
)
=
h
(
x
)
=
n
(
x
)
=
,
,
and
.
x+2
x 2 − 2x − 3
x2 + 4
x2 − 4
Explain each of the following.
a) Which of these rational functions has a horizontal asymptote?
b) Which has an oblique asymptote?
c) Which has no vertical asymptote?
d) Graph y = n(x), showing the asymptote and x- and y- intercepts.
10. A farmer wants to experiment with a new wheat plant. He has enough seed for an area of 216
m2. He wants to fence a rectangular plot. This plot will be divided into two equal parts by
another fence parallel to one side. What dimensions for the largest rectangle will minimize
the amount of fencing.
The area is 216 m2. Now express the length of the largest rectangle in terms of x.
Express the perimeter in terms of x.
Describe the relation between the perimeter and x.
Enter the perimeter function into a graphing calculator. Describe the graph. What is
the domain of the perimeter function in the context of this problem?
e) Find the smallest possible perimeter using the graphing calculator.
f) How much fencing will the farmer need?
a)
b)
c)
d)
11. Juicy Drinks wants to construct a closed rectangular container with a square base. The
container must hold 0.2 L (200 cm3). Let S be the surface area of the box and x be the length
of a side of the square base. Show that S ( x ) = 2 x 2 +
800
for x>0. Graph S versus x for
x
x>0. Estimate the dimensions of the container that will require the least amount of material
for its manufacture.
Answers 1.a) max is approximately 13, increasing –3 ≤x≤.5, deceasing x>.5 or x< -3, 3. domain is 0 ≤ x ≤ 4
4. h(x) = 4x4 + 2x3 –2x2 – 13x – 3 5. D(t) = (10t+5)(2t), 2100 m 6.a) l(t) = 100 + 20t b) w(t) = 50 + 10t
c) A(t) = (100+20t)(50+10t) = 200t2 + 2000t+5000 d) A(2010)=45000 7) f(x) = 3x2 + 2x – 5 and
g(x) = 2x2 – 4x - 2 8. y=3/(x2-1) 9.a) f and n b) g c) g d) x-intercept = -4 or 3, y-intercept = 3, V.A. x = 2
and x = -2, H.A. y = 1 10.a) length = 216/x b) p(x)=(432+3x2)/x c) x>0 d) P = 72 m 11) x=5.85cm (cube)
6.3B – combining functions (multiplying and dividing)