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

1.3 Creating New Polynomial 

Functions: An Introduction 

to Composition 

SETTING THE STAGE 

EXAMINING THE CONCEPT 

Can you combine polynomial functions using the basic mathematical 

operations If so, what is the result 

Rhona is a licensed plumber and Bill is an apprentice training for his licence. 

Rhona earns $28/h and receives $22/day to travel to the job site. Bill earns $12/h 

and receives a $10/day travel allowance. The polynomial model for Rhona’s 

daily earnings, R, in dollars is R(h) 28h 22, where h is the number of hours 

worked. Similarly, Bill’s earnings are modelled by B(h) 12h 10. 

In this section, you will examine what happens when polynomial functions 

are combined using familiar operations to form new polynomial functions. 

Adding and Subtracting Functions 

When Rhona and Bill are working together on a job, the contractor for the job 

site needs a single function to determine the costs for plumbing work. 

The total daily cost, T(h), is a function of the number of hours worked, h, by 

Rhona and Bill. 

Rhona’s Bill’s Total Earnings, 

Time Earnings, R(h) Earnings, B(h) T(h) R(h) B(h) 

(h) ($) ($) ($) 

0 22 10 22 10 32 

1 50 22 50 22 72 

2 78 34 78 34 112 

3 106 46 106 46 152 

So 

T(h) R(h) B(h) 

(28h 22) (12h 10) 

28h 22 12h 10 

40h 32 

In both the table and the graph, the 

values of the functions are added 

together to obtain the value of the 

new function. 

Adding or subtracting two 

functions means adding or 

subtracting the corresponding 

outputs, or, in terms of the graph, 

adding or subtracting the 

y-coordinates. 

Earnings ($) 

160 

140 

120 

100 

80 

60 

40 

20 

Total cost to 

contractor 

y = 40h + 32 

0 1 2 3 

Time (h) 

(3, 152) 

(152 = 46 + 106) 

Rhona's earnings 

y = 28h + 22 

Bill's earnings 

y = 12h + 10 

28 CHAPTER 1 POLYNOMIAL FUNCTION MODELS

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


Multiplying Two Polynomial Functions 

What is the result when two polynomial functions are multiplied 

Example 2 

Multiplying Two Polynomial Functions 

Let f (x) x 3 and g(x) x 2 5, x ∈ R. 

(a) Sketch each graph on the same axes. 

(b) Make a table for 3 ≤ x ≤ 3, and determine the corresponding values of 

h(x) f (x) g(x). 

(c) Use the table to sketch h(x) on the same axes. Describe the shape of the 

graph. 

(d) Determine the algebraic model for h(x). What is its degree 

(e) What is the domain of h(x) How does this domain compare with the 

domains of f (x) and g(x) 

10 

8 

6 

4 

2 

–5 –4 –3 –2 –1 

–2 

y = h(x) 

18 

15 

12 

9 

6 

y 

–4 

–6 

–8 

–10 

3 

x 

–3 –2 –1 1 2 3 

–3 

–6 

y = g(x) 

–9 

y 

Solution 

y = f(x) 

1 2 3 4 5 

y = f(x) 

y = g(x) 

x 

(a) Use intercepts to sketch the graphs. 

The graph of function f (x) is linear with slope 1 and 

y-intercept 3. 

The graph of function g(x) is quadratic with vertex (0, 5). 

The parabola opens down. 

(b) 

f (x) x 3 g(x) x 2 5 h(x) f (x) g(x) 

3 0 4 0 (4) 0 

2 1 1 1 1 1 

1 2 4 2 4 8 

0 3 5 3 5 15 

1 4 4 4 4 16 

2 5 1 5 1 5 

3 6 4 6 (4) 24 

(c) The graph of h(x) is a cubic polynomial. 

(d) h(x) f (x) g(x) 

(x 3)(x 2 5) 

x 3 3x 2 5x 15 

This polynomial function has degree 3. 

(e) The domain of h(x) is the set of real numbers, which is the same as the 

domains of f (x) and g(x). 


Adding, Subtracting, and Multiplying 

Polynomial Functions 

For the polynomial functions f and g, 

• the sum is ( f g)(x) f (x) g(x) 

• the difference is ( f g)(x) f (x) g(x) 

• the product is ( fg)(x) f (x) g(x) 

The domain of f ± g and fg is the set of all x, for which both f and g are 

defined. Any x belonging to the domain of f and to the domain of g belongs 

to the domain of f (x) ± g(x) or f (x) g(x). 


input → output 

x → g(x) → f (g(x)) 

g f 

input → output 

The Composition of Functions 

What happens when the output of one function is the input of another 

You can think of a function as an input-output machine. The input set 

is the domain of the function, and the output set is the range of the function. 

In the first step of the diagram, the x is an input and g(x) is an output. 

In the second step, g(x) is an input and f (g(x)) is an output. 

Combining two functions in this way is called composition. Composition is 

the result of substituting the output of one function into another. The new 

function is called the composite of f and g and is written f g. 

Example 3 

Determining the Equation of a Composite Function 

Recall the situation in Setting the Stage. Rhona earns a daily wage according to 

R(h) 28h 22. The job site where Rhona works requires all employees to 

become members of a union. Rhona then pays 1.5% of her daily earnings as 

union dues. Determine the function that represents her daily union dues. 

Hours Worked 

h 

1 

2 

3 

. 

. 

. 

. 

8 

R 

Daily Earnings 

R(h) 

50 

78 

106 

. 

. 

. 

. 

246 

Solution 

U 

Union Dues 

U(R(h)) 

0.75 

1.17 

1.59 

. 

. 

. 

. 

3.69 

This situation involves two functions, 

Daily Earnings: R(h) 28h 22 

Daily Union Dues: U(R(h)) 0.015 R(h) 

Examine the mapping that shows how the number of 

hours worked per day is related to the daily union dues. 

The domain for Rhona’s daily earnings is the number of 

hours worked. But the domain for the daily union dues 

are her daily earnings, not the number of hours worked. 

In other words, the domain of the substitution function is 

the range of the substituted function. 


Find the composite of the two functions. 

U(R(h)) 0.015(R(h)) 

0.015(28h 22) 

0.42h 0.33 

The function U(R(h)) 0.42h 0.33 represents Rhona’s daily union dues. 

Composite Functions 

For two functions f and g, the composite function f (g(x)) is formed by 

evaluating f at g(x). This new function is defined as f g. The domain of f g 

is a subset of the domain of g and is found by examining f g and comparing 

the domain of f with the range of g. 

Example 4 

Determining the Value of a Composite Function 

For the functions f {(4, 1), (3, 2), (2, 3)} and g {(1, 5), (2, 4), (4, 2)}, 

determine each value. 

(a) f (g(4)) (b) g( f (4)) 

Solution 

(a) First find the value of g where x 4. In the ordered pair (4, 2), x 4. 

Therefore, g(4) 2. 

Then f (g(4)) f (2). Now find the value of f where x 2. 

In the ordered pair (2, 3), x 2. 

So f (g(4)) f (2) 

3 

(b) Find the value of f where x 4. In the ordered pair (4, 1), x 4. 

Therefore, f (4) 1. 

Then g( f (4)) g(1). Now find the value of g where x 1. 

In the ordered pair (1, 5), x 1. 

So 

g( f (4)) g(1) 

5 

Example 5 

Applying a Composite Function 

Ali buys a pair of shoes and has a coupon for $15 off. The day he buys the 

shoes, the store has a sale offering 20% off the price of all shoes. 

(a) Write a function to represent the cost, C, of a pair of shoes at price, p, if just 

the coupon reduction is applied to the cost of the shoes. Write a function that 

represents the cost, D, if just the discount is applied to the cost. 


(b) What does the function D(C(p)) represent if a pair of shoes has a regular 

price of $80 Determine D(C(80)). 

(c) What does the function C D represent for a pair of shoes with a regular 

price of $80 Determine C D for the same pair of shoes. 

(d) Does C D D C Justify your answer. 

Solution 

(a) C(p) p 15 and D(p) 0.8p 

(b) D(C(p)) means the coupon is applied first and then the 

discount is applied to the cost of the shoes. So, when p 80, 

D(C(80)) D(80 15) 

D(65) 

0.8(65) 

52 

(c) C D means the discount is taken first and then the coupon reduction is 

applied. When p 80, 

(C D)(x) C(D(80)) 

C(0.8(80)) 

C(64) 

64 15 

49 

(d) In this case, C D ≠ D C. The order of composition makes a difference. 

When p 80, D(C(80)) 52 and C(D(80)) 49. In fact, these two 

functions are different. 

(C D)(p) C(0.8p) (D C)(p) D(p 15) 

0.8p 15 0.8(p 15) 

0.8p 12 

CHECK, CONSOLIDATE, COMMUNICATE 

1. Explain how a new polynomial function can be made from two 

polynomial functions by addition, by subtraction, and by multiplication. 

2. State ( f g)(x), ( f g)(x), ( f g)(x), and f (g(x)) for 

f {(1, 4), (0, 5), (6, 3)} and g {(1, 6), (0, 9), (4, 2)}. 

3. Determine ( f g)(x), ( f g)(x), ( f g)(x), and f (g(x)) for 

f (x) x 2 3x 2 and g(x) x 3. 


KEY IDEAS 

• New polynomial functions can be made from other polynomial functions 

through addition, subtraction, multiplication, and composition. 

• The domain of a polynomial function that is the result of adding, 

subtracting, or multiplying other polynomial functions is the same as the 

common domain of the original polynomial functions. 

• For two functions f and g, the composite function f g f (g(x)) is formed 

by evaluating f at g. For f g, the domain of f is the range of g, provided 

that g is defined. 

1.3 Exercises 

A 

1. Knowledge and Understanding: Let f {(4, 4), (2, 4), (1, 3), (3, 5), 

(4, 6)} and g {(4, 2), (2, 1), (0, 2), (1, 2), (2, 2), (4, 4)}. Determine 

(a) f g (b) f g (c) g f (d) fg 

2. Use the graph of f and g to sketch the graph of f g. 

(a) 

y 

(b) 

f 

7 

6 

5 

4 

3 

2 

g 1 

x 

–5 –4 –3 –2 –1 0 

–1 

1 2 3 4 5 

–2 

f 

7 

6 

5 

4 

3 

2 

1 

–5 –4 –3 –2 –1 0 1 2 3 4 5 

–1 

–2 

y 

g 

x 

3. Use the graph of f and g to sketch the graph of f g. 

(a) 

y 

7 

6 

5 

4 

f 3 

2 

1 

x 

–5 –4 –3 –2 –1 0 1 2 3 4 5 

g –1 

–2 

(b) 

y 

7 

6 

f 

5 

4 

3 g 

2 

1 

–5 –4 –3 –2 –1 0 

–1 

1 2 3 4 5 

–2 

x 


4. Use the graph of f and g to sketch the graph of fg. 

(a) 

y 

7 

6 

f 

5 

4 

g 

3 

(b) 

2 

1 

x 

–3 –2 –1 0 

–1 

1 2 3 4 5 6 7 

–4 –3 

–2 

y 

7 

g 6 

5 

4 

3 

2 

1 

–2 –1 0 

–1 

1 2 

–2 

3 

f 

4 

x 

B 

5. Let f {(3, 2), (5, 1), (7, 4), (9, 3), (11, 5)} and 

g {(1, 3), (2, 5), (3, 7), (4, 9), (5, 11)}. Determine 

(a) f (g(3)) (b) g( f (9)) 

6. Let f {(1, 3), (0, 2), (1, 3), (2, 6)} and 

g {(1, 0), (0, 1), (1, 2), (2, 3)}. Determine 

(a) f (g(x)) 

(b) g( f (x)) 

7. Let f (x) 3x 2 and g(x) 2x 4, x ∈ R. Determine an expression for 

each case, and state the domain of the expression. 

(a) ( f g)(x) (b) (g f )(x) (c) ( f g)(x) (d) ( fg)(x) 

8. Let f (x) 3x 1, 0 ≤ x ≤ 6, and g(x) x 2 6x, 1 ≤ x ≤ 4. 

Determine an expression for each case, and state the domain of 

the expression. 

(a) ( f g)(x) (b) ( f g)(x) (c) (g f )(x) (d) ( fg)(x) 

9. Let f (x) x 2 4 and g(x) 3x. Determine 

(a) f (g(1)) (b) g( f (2)) (c) g(g(1)) (d) f ( f (2)) 

10. Communication: Consider two functions f and g. 

(a) Explain why the domains of f and g must be the same in order to add, 

subtract, or multiply the functions. 

(b) Explain why the domain of f is the range of g for f (g(x)). 

11. Let f {(1, 4), (0, 2), (1, 0), (2, 2)}. 

(a) Give an example of g if f g {(1, 6), (0, 4), (1, 2), ( 2, 0)}. 

(b) Give an example of g if fg {(1, 12), (0, 6), (1, 0), (2, 6)}. 

12. Let f (x) 3x 2, 2 ≤ x ≤ 4, and g(x) 4x 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. 

What is the domain of f g 


13. In each case, determine f (x) g(x) and draw its graph. 

(a) f (x) 2x 3, x ∈ R, and g(x) 3x 4, x ∈ R 

(b) f (x) 3x 2, x ∈ R, and g(x) x 2 3, x ∈ R 

(c) f (x) x 2 2, x ∈ R, and g(x) 2x 2 1, x ∈ R 

14. In each case, functions f and g are defined for x ∈ R. For each pair of 

functions, determine the expression and the domain for f (g(x)) and g( f (x)). 

Graph each result. 

(a) f (x) 3x 2 , g(x) x 1 

(b) f (x) 2x 2 x, g(x) x 2 1 

(c) f (x) 2x 3 3x 2 x 1, g(x) 2x 1 

(d) f (x) x 4 x 2 , g(x) x 1 

15. In each case, x ∈ R for f (x) and g(x). Graph f (x) and g(x) on the same axes 

and use this graph to graph h(x) f (x) g(x). 

(a) f (x) x 2 1 and g(x) 3 

(b) f (x) 2x 2 2 and g(x) 2x 2 1 

(c) f (x) (x 1)(x 2)(x 4) and g(x) (x 1)(x 4)(x 6) 

16. Verify your work in question 15 using a graphing calculator. 

17. Repeat question 15 for h(x) f (x) g(x). 

18. A company produces a product for $9.45 per unit, plus a fixed operating 

cost of $52 000. The company sells the product for $15.80 per unit. 

(a) Determine a function, C(x), to represent the cost of producing x units. 

(b) Determine a function, S(x), to represent sales of x units. 

(c) Determine a function that represents profit. 

19. Steve earns $24.39/h operating an industrial plasma torch at a rail car 

manufacturing plant. He receives $0.58/h more for working the night shift, 

as well as $0.39/h for working weekends. 

(a) Write a function that describes Steve’s regular pay. 

(b) What function shows his night shift premium 

(c) What function shows his weekend premium 

(d) What function represents his earnings for the night shift on Saturday 

(e) How much does Steve earn for working 11 h on Saturday night, if he 

earns time and a half on that day’s rate for more than 8 h of work 

20. A circle has radius r. 

(a) Write a function for the circle’s area in terms of r. 

(b) Write a function for the radius in terms of the circumference, C. 

(c) Determine A(r(C)). 

(d) A tree’s circumference is 3.6 m. What is the area of the cross section 


Johann Peter Gustav 

Lejeune Dirichlet 

(1805–1859) 

Originally, the 

definition of a function 

was much less precise 

than it is today. 

Dirichlet was 

responsible for the 

definition of a function 

that we use today. 

21. Application: A solar panel is used to power an overhead sign on a highway. 

Each square metre of solar panel receives about 200 W of solar power. This 

solar panel converts about 15% of the solar energy to electrical power. 

(a) Write a function for the solar power, S(A), measured in watts (W) for 

any given area, A, in square metres. 

(b) Write a function for the total electrical power, P(S), in watts (W) this 

solar panel can generate. 

(c) How large must the solar panel be if the sign uses 6 W of electrical 

energy 

22. Refrigeration slows down the growth of bacteria in food. The number of 

bacteria in a certain food is approximated by B(T) 15T 2 70T 600, 

where T represents the temperature in degrees Celsius and 3 ≤ T ≤ 12. 

Once the food is removed from refrigeration, the temperature, T(t), 

is given by T(t) 3.5t 3, where t is the time in hours and 0 ≤ t ≤ 3. 

(a) Write the expression for the number of bacteria in the food, t hours 

after it is removed from refrigeration. 

(b) At 1.5 h, about how many bacteria are in the food 

(c) When will the bacteria count reach about 1200 

23. A franchise owner operates two coffee shops. The sales, S 1 

, in thousands of 

dollars, for shop one are represented by S 1 

(t) 700 1.4t 2 , where t 0 

corresponds to the year 2000. Similarly, the sales for shop two are 

represented by S 2 

(t) t 3 3t 2 500. 

(a) Which shop is showing an increase in sales after the year 2000 

(b) Determine a function that represents the total sales for the two 

coffee shops. 

(c) What are the expected total sales for the year 2006 

(d) If sales continue according to the individual functions, what would you 

recommend that the owner do Explain. 

C 

24. Thinking, Inquiry, Problem Solving: Let f (x) mx 2 2x 5 and 

g(x) 2x 2 nx 2. The functions are combined to form the new function 

h(x) f (x) g(x). Points (1, 40) and (1, 24) satisfy the new function. 

Determine f (x) and g(x). 

25. Let h(k) 3k 2, g(t) 3t 2, and f (x) 3x 2. 

(a) Determine C(k) f (g(h(k))). 

(b) Show in a diagram how k 4 carries through to its final value. 


ADDITIONAL ACHIEVEMENT CHART QUESTIONS 

Knowledge and Understanding: Let f (x) x 2 3x 1 and g(x) 2x 5. 

Determine 

(a) f (x) g(x) (b) f (x) g(x) (c) f (x) g(x) (d) f (g(x)) 

Application: The temperature of the Earth’s crust is a linear function of the depth 

below the surface. The function T(d) 0.01d 20 gives the temperature, T(d), 

in degrees Celsius d metres below the Earth’s surface. An elevator goes down a 

mine shaft at 7.5 m/s. Express the temperature as a function of the time 

travelled. 

Amount ($1000s) 

990 

900 

810 

720 

630 

540 

450 

360 

270 

180 

Income 

Expense 

Profit 

Thinking, Inquiry, Problem Solving: If f (x) ax b, 

g(x) cx d, and f g(x) g f (x), determine the relation 

among a, b, c, and d. 

Communication: The graph on the left shows two functions and 

the function that represents the difference between the two 

functions. Identify and describe the two original functions and 

the difference function. 

90 

–90 

–180 

1 2 3 4 5 6 7 8 9 10 

Number of Units (1000s) 

The Chapter Problem 

Developing a Model for Canada’s Population 

In this section, you learned about creating new polynomial functions. 

Apply what you learned to answer these questions about The Chapter 

Problem on page 2. 

CP9. 

Refer to the two different polynomial models that you created for 

question CP5 (section 1.2). Write the linear and quadratic 

functions to model the population of Canada. 

CP10. Graph both functions on the same axes. 

CP11. Determine the polynomial function that represents the average of 

the previous models, and graph the new function. 

CP12. Use the new function to estimate the population of Canada in 

2001, 2006, and 2016. Compare your results with other data on 

the Internet. How do they compare

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?