6.3A – Combining Functions (Adding and Subtracting)
Combining functions refers to adding or multiplying two or more functions to create a new
function. This might involve combining functions that use the same variable or functions that use
different variables. Although it might be preferred to keep equations as functions of just one
variable this is not always possible with combined function.
Ex.
Example 1:
If g(x) = 2x + 3 and f(x) = x – 5 then might want m(x) = g(x) + f(x)
m(x) = (2x + 3) + (x – 5)
m(x) = 3x – 2
You have just been promoted to contractor for a major renovating company. You
need to determine the cost for some electrical work on a proposed job. You have subcontracted to Shocks ’R’ Us Electrical. Jackie is a licensed electrician and Bill is an
apprentice training for his licence. Jackie early $30/h and received $40/day to travel
to the job site. Bill earns $14/h and received a $20/day travel allowance.
a) Determine a linear function to represent Jackie’s earnings:
J(h) =
b) Determine a linear function to represent Bill’s earnings:
B(h) =
c) Complete the table of values for earnings for a work day of up to 8 hours.
d) Sketch both functions on the grid below.
e) Fill in the last column of the table for Total earnings by adding Jackie’s earnings and
Bill’s earnings.
f) Graph the Total Earnings on the grid above
g) Create a function to represent the Total Earnings for Jackie and Bill. T(h) =
h) How could you have determined the function for Total Earnings without graphing?
Time (h)
0
1
2
3
4
5
6
7
6.3A – combining functions (adding and subtracting)
Jackie’s
Earnings ($)
Bill’s Earnings
($)
Total
Earnings ($)
40
70
100
20
34
60
Example 2:
Let f(x) = 2x – 3 and g(x) = 2x + 1, x∈R. Determine an expression for each case,
and state the domain of the expression.
a) (f + g)(x)
b) g(x) – f(x)
c) (f - g)(x)
d) 3f(x) + 2g(x)
= (2x – 3) + (2x + 1)
= 2x – 3 + 2x + 1
= 4x – 2
= (2x + 1) – (2x – 3)
= 2x + 1 – 2x + 3
=4
= (2x – 3) – (2x + 1)
= 2x – 3 – 2x - 1
=-4
= 3(2x-3) + 2(2x+1)
= 6x – 9 + 4x + 1
= 10x - 8
Example 3:
Use the graph of f and g to sketch the graph of f + g and f – g.
f(x)
Example 4:
g(x)
f(x)
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. Approximate
maximum and minimum points.
b) Graph h(x) = f(x) – g(x)
c) Identify the domain of each function.
d) Determine when each function is increasing, decreasing, and zeroes.
e) Approximate the value(s) of the maximum and minimum of the new function.
f) Verify approximations with a graphing calculator.
6.3A – combining functions (adding and subtracting)
g(x)
6.3A – Combining Function (Adding and Subtracting) Practice Questions
1. Let f={(-4,4),(-2,4),(1,3),(3,5),(4,6)} and g = {(-4,2).(0,2),(1,2),(2,2),(4,4)}. Determine
a) f + g
b)f – g
c) g – f
2. Use the graph of f and g to sketch the graph of f(x) + g(x) and f(x) – g(x).
a)
f(x)
b)
f(x)
g(x)
g(x)
3. If a linear and quadratic function are added what would be the overall degree of the new
function?
4. Let f(x) = 3x – 2, -2 ≤ x ≤ 4 and g(x) = -3x + 1, -3 ≤ x ≤ 5.
a) Graph f and g on the same axes
b) Use appropriate ordered pairs to sketch the graph of f – g.
c) What is the domain of f – g?
5. Let f(x) = 3x – 2 and g(x) = 2x + 1, x∈R. Determine an expression for each case, and sate
the domain of the expression.
a) (f+g)(x)
b) (g-f)(x)
c) (f-g)(x)
6. Apple produces an I-pod for $20 per unit, plus a fixed operating cost of $250 000. The
company sells the product for $99.99 per unit.
b) Determine a function, C(x), to represent the cost of producing x units.
c) Determine a function, S(x), to represent sales of x units.
d) Determine a function that represents profit.
7. Let f(x) = 3x – 1, 0<x<6 and g(x) = x2 – 6x, -1<x<4.
a) Graph f and g on the same axes
b) Use appropriate ordered pairs to sketch the graph of f – g.
c) What is the domain of f – g?
8. Let f(x) = 2x – 3 and g(x) = 2x3 + 4x2 + 5x + 1, x∈R. Determine an expression for each case,
and sate the domain of the expression.
a) (f+g)(x)
6.3A – combining functions (adding and subtracting)
b) (g-f)(x)
c) (f-g)(x)
9. A franchise owner operates two Tim Horton’s. The sales, S1, in thousands of dollars, for
Timmy’s # 1 are represented by S1(t) = 800 – 2t2, where t = 0 corresponds to the year 2005.
Similarly, the sales for Timmy’s # 2 are represented by S2(t) = t3 + 3t2 + 500.
a)
b)
c)
d)
Which Timmy’s is showing an increase in sales after 2005?
Determine a function that represents the total sales for the two Timmy’s.
What are the expected total sales for the year 2007?
If sales continue according to the individual functions, what would you recommend that
the owner do? Explain
10. Describe how to determine the type of function for (f+g)(x) if we know the degree of f(x)
and the degree of g(x).
11. Bob earns $19.45/h operating a forklift at Home Depot. He receives $0.64/h more for
working the evening shift, as well as $0.39/h for working weekends.
a)
b)
c)
d)
e)
Write a function that describes Bob’s regular pay.
What function shows his evening shift premium?
What function shows his weekend premium?
What function represents his earnings for the evening shift on Saturday?
How much does Bob earn working 11 h on Saturday evening, if he earns time and half
on that day’s rate for more than 8 hours of work?
12. Consider two functions f and g. Explain why the domains of f and g must be the same in
order to add or subtract the functions.
13. Let f(x) = mx2 + 2x + 5 and g(x) = 2x2 – nx – 2. The functions are combined to form the new
function h(x) = f(x) + g(x). Points (1, 6 ) and (-1,10) satisfy the new function. Determine
f(x) and g(x)
14. Investigate the effect of the behaviours of f(x) = sin x, f(x) = sin2x, and f(x) = sin4x on the
shape of f(x) = sin x + sin2x + sin4x.
Answers:1.a) {(-4,6),(1,5),(4,10)} b) {-4,2),(1,1),(4,2)} c) {(-4,2),(1,-1),(4,-2)} 3. 2nd degree (quadratic)
4. D: -2 ≤ x ≤ 4 5.a) (f+g)(x) = 5x – 1 b) (g-f)(x) = -1x + 3 c) (f-g)(x) = 1x – 3 6.a) C(x) = 20x + 250 000
b) S(x) = 99.99x c) P(x) = S(x) – C(x) = 79.99x – 250 000 7. D: 0 < x < 4 8.a) 2x3+4x2+7x-2
b) 2x3+4x2+3x+4 c) -2x3-4x2-3x-4 9.a) graph shows Timmy’s #2 b) s(t)=t3+t2+1300 c) s(2007)=$1312000
d) Sell Timmy’s #1 in2025. Its sales go to zero then. Might sell earlier based on point when profit goes to
zero l0. Shape is determined by highest degree function 11.a) B(t)=19.45t b) E(t)=0.64t c) w(t)=0.39t
d) P(t)=B(t)+E(t)+W(t) = 20.48t e) Pay=20.48(8)+ 20.48(1.5) 12. Because one cannot to a number that does
not exist (i.e. if domain is not defined for a portion you want to combine)
13. f(x)=3x2+2x+5 and g(x)= 2x2+4x-2 14. Graphing calculator shows max of 2 and minimum of -2 with
period around 7 radians (about 2π) with many little fluctuation in the graph.
6.3A – combining functions (adding and subtracting)