Functions
and Graphs
Learning Outcomes
In this chapter you will learn to:
 Understand the concept of a function
 Recognise couples, domain, codomain and range
 Write functions using different notations
 Draw graphs of linear functions
 Draw graphs of quadratic functions
 Use graphs to solve problems
Functions and graphs
1 ide a oF a Function
A function is like a machine. If you put in a number, the machine
changes the number and sends out a new number.
For example, let us take set A = {4, 10, 11.5}.
Let f be a function which doubles the input.
If you put in 4, then 8 will come out. We write
f(4) = 8. If the input is 10, the output will be 20.
This can be written f(10) = 20. If you send in
11.5, 23 will emerge. In other words f(11.5) = 23.
The domain of this function is {4, 10, 11.5},
the set of inputs.
The range of this function is {8, 20, 23},
the set of outputs.
A function which doubles numbers could
be represented in many other ways:
f(x) = 2x
y = 2x
f : x → 2x
It is important to note that the variable x is usually
used to represent the input.
y or f(x) are used to represent the corresponding output.
A function can also be written as a set of couples. The input is written as the first component,
and the output is written as the second component. For example, the function f could be written:
f = {(4,8), (10,20), (11.5,23)}
When a function is written as a set of couples, all the first components will be different.
2 do main and r ange
Wo rked exa m p le 1
f : x → 3x – 1 is a function defined on the domain {1, 2, 3, 4}. Find the range.
Solution
‘f : x → 3x – 1’ means ‘If you put in a number, you will get out three-times-the-number-minusone’. See what happens to the members of the domain 1, 2, 3 and 4:
f(1) = 3(1) – 1 = 3 – 1 = 2
(If you put in 1, you get out 2.)
f(2) = 3(2) – 1 = 6 – 1 = 5
(If you put in 2, you get out 5.)
f(3) = 3(3) – 1 = 9 – 1 = 8
(If you put in 3, you get out 8.)
f(4) = 3(4) – 1 = 12 – 1 = 11 (If you put in 4, you get out 11.)
∴ f = {(1,2), (2,5), (3,8), (4,11)}
∴ The range of this function is {2, 5, 8, 11} (the set of outputs).
2
Worked exa m p le 2
Functions and graphs
g is the function g : x → 5x + 4. Find the values of v, w and x, as in the diagram.
Solution
g(2) = 5(2) + 4 = 10 + 4 = 14 ∴ v = 14
() ()
1
1
1
1
g __ = 5 __ + 4 = 2__ + 4 = 6__
2
2
2
2
g(x) = 5x + 4 = 39
1
∴ w = 6__
2
∴ 5x = 35
(taking 4 from both sides)
∴x=7
(dividing both sides by 5)
codomain
Let A = {–1, 0, 1}
Let B = {0, 1, 2}
Let f : A → B : x → x2 + 1
This means that f is a function which transforms elements of the set A into elements of the set B,
and the output is the input squared plus one.
A is the domain. B is the codomain, the set of allowable outputs.
f(–1) = (–1)2 + 1 = 1 + 1 = 2
f(0) = (0)2 + 1 = 0 + 1 = 1
f(1) = (1)2 +1 = 1 + 1 = 2
∴ f = {(–1,2), (0,1), (1,2)}
A = {–1, 0, 1} = the domain.
B = {0, 1, 2} = the codomain = the set of allowable outputs.
{1, 2} = the range = the set of actual outputs.
Life is full of functions. For example, here is
a function which shows the average seasonal
temperatures (in degrees Fahrenheit) in Alaska:
Here is another function which shows the
prices of various items in a local grocery store:
Here is a function which tells you the code
to dial if you want to telephone someone in
a particular place:
3
Functions and graphs
Finally, f : x → 20 + 45x is a function in which the input is the mass of
a roast beef (in kilograms) and the output is the time (in minutes) which
the roast should spend in the oven.
exercise 1
1. f : x → 5x + 2 is a function. Find f(3).
9. f is a function such that f(x) = 10 – x.
2. g : x → 7x – 1 is a function. Evaluate g(2).
(i) Find f(6).
3. f : x → 10x + 3. Find f(7).
(ii) If the domain of f is {6, 7, 8, 9, 10, 11},
find the range.
4. g : x → 6x – 10. Find g(5).
5. f : x → 2x + 1 is a function.
10. f : x → x2 + 1 is a function defined on the
domain {–2, –1, 0, 1, 2}. Find the range.
11. f : x → 3x + 11 is a function.
The domain is {1, 2, 3}. Find the range.
12. f : x → 7x – 10 is a function.
The domain is {0, 1, 2, 3}. Find the range.
Copy and complete the diagram, replacing the
question marks with the appropriate values.
6. g : x → 4x – 3 is a function.
13. g : x → x2 + 4 is a function.
The domain is {–5, 0, 5}. Find the range.
14. f : x → 2x2 is a function. Evaluate:
(i) f(3)
(ii) f(5)
(iii) f(10)
15. f : x → 3x2 – 5 is a function.
If the domain is {2, 3, 4, 5}, find the range.
16. f : x → 10x2 – 3 is a function. Evaluate:
Write down the values of a, b and c, as in the
diagram.
7. f : x → 9 – x is a function, as shown in the
diagram.
(i) f(1)
(ii) f(2)
(iii) f(3)
17. f : x → 5x – 8 is a function. Find:
(i) f(7)
(ii) f(–7)
(iii) The value of p if f(p) = 17
(iv) The range of q if f(q) = q
18. g : x → 2x + 3 is a function defined on the
domain N (the set of natural numbers).
(i) Find the values of g(1), g(2) and g(3).
Write down the values of p, q, r and s.
8. f : x → 5x – 2 is a function. The domain of f is
{1, 2, 3}. Find the range of f.
4
(ii) Investigate if g(3) = g(1) + g(2).
(iii) Find the value of x if g(x) = 53.
6. g : x → 4x – 3 is a function.
Functions and graphs
Copy and complete the diagram, replacing
the question marks with the appropriate
values.
Write down the values of p, q, r and s.
8. f : x → 5x – 2 is a function. The domain of f is
{1, 2, 3}. Find the range of f.
9. f is a function such that f(x) = 10 – x.
(i) Find f(6).
(ii) If the domain of f is {6, 7, 8, 9, 10, 11},
find the range.
Write down the values of a, b and c, as in the
diagram.
7. f : x → 9 – x is a function, as shown in the
diagram.
10. f : x → x2 + 1 is a function defined on the
domain {–2, –1, 0, 1, 2}. Find the range.
11. f : x → 3x + 11 is a function.
The domain is {1, 2, 3}. Find the range.
12. f : x → 7x – 10 is a function.
The domain is {0, 1, 2, 3}. Find the range.
13. g : x → x2 + 4 is a function.
The domain is {–5, 0, 5}. Find the range.
14. f : x → 2x2 is a function. Evaluate:
5
Functions and graphs
graph of a Linear Function
Wo rke d exa m p le 3
f : x → 2x – 1 is a function.
(i) Draw the graph of f in the domain –1 ≤ x ≤ 4.
(ii) Estimate from the graph the value of f(2.2).
(iii) If f(x) = 1.8, find the value of x.
Solution
(i) The domain (the set of inputs) of this function is {–1, 0, 1, 2, 3, 4}.
The couples of f will be {(–1,–3), (0,–1), (1,1), (2,3), (3,5), (4,7)}.
If we plot these points, we see that they form a straight line.
By joining these points we form the graph of this function.
(ii) This means: ‘If the input is 2.2, find the output.’ Draw a dotted line
from 2.2 on the x-axis to the graph and then to the y-axis.
Answer: 3.4
(iii) This means: ‘If the output is 1.8, find the input.’ Draw a dotted line
from 1.8 on the y-axis to the graph and then to the x-axis.
Answer: 1.4
Worke d e xa m p le 4
Solve graphically the simultaneous equations y = 4 – x and y = 2x – 5,
by graphing both functions in the domain 0 ≤ x ≤ 5.
Solution
Here are the couples of the first function:
{(0,4), (1,3), (2,2), (3,1), (4,0), (5,–1)}
Here are the couples of the second function:
{(0, –5), (1, –3), (2, –1), (3,1), (4,3), (5,5)}
On the right we see the graphs of the two functions simultaneously.
The point of intersection is (3,1), which is the only point which
satisfies both equations simultaneously
Answer: x = 3, y = 1
6
exercise 2
2. Draw the graph of the function:
y = 3x + 1 in the domain –1 ≤ x ≤ 5.
3. (i) Draw the graph of the function:
f : x → 3x – 3 in the domain 0 ≤ x ≤ 5.
(ii) From your graph, estimate the value of
f(3.3).
4. (a) Draw the graph of the function:
f : x → 2x + 3 in the domain –2 ≤ x ≤ 4.
(b) Estimate (from your graph):
(i) The value of f(1.5)
(ii) The value of x for which f(x) = 8
5. (a) Draw the graph of the function:
y = 5 – x in the domain –1 ≤ x ≤ 4.
(b) Estimate from your graph:
(i) The value of y if x = 3.3
(ii) The value of x if y = 5.4
6. (a) Draw the graph of the function:
y = 6 – 2x in the domain –2 ≤ x ≤ 4.
(b) Estimate from your graph:
(i) The value of y if x = 1.3
(ii) The value of x if y = 1.4
7. (a) Draw the graph of the function:
f : x → 4x – 5 in the domain –2 ≤ x ≤ 3.
(b) Estimate from your graph:
Functions and graphs
1. Draw the graph of the function:
f : x → 2x + 1 in the domain –1 ≤ x ≤ 4.
(i) The value of f(0.2)
(ii) The value of x if f(x) = 6
8. (i) Using the same scales and axes, draw the
graphs of the two functions y = 3 – x and
y = x – 1 in the domain 0 ≤ x ≤ 4.
(ii) Use your graphs to solve the simultaneous
equations y = 3 – x and y = x – 1.
9. (i) Using the same scales and axes, draw the
graphs of the two functions y = –2x and
y = 3x – 5 in the domain –1 ≤ x ≤ 3.
(ii) Use your graphs to solve the simultaneous
equations y = –2x and y = 3x – 5.
10. (i) Using the same scales and axes, draw (in the
domain –3 ≤ x ≤ 1) the graphs of the two
functions y = –x and y = 2x + 3.
(ii) Use your graphs to solve the simultaneous
equations y = –x and y = 2x + 3.
11. (i) Using the same scales and axes, draw
the graphs of these two functions in the
domain 0 ≤ x ≤ 5:
f : x → __12 x – 3 and g : x → 3 – x
(ii) Use your graphs to solve the simultaneous
equations y = _12 x – 3 and y = 3 – x.
12. (a) Solve the simultaneous equations y = 2 – x
and y = 3x – 10 using an algebraic method.
(ii) Solve these equations by graphing the
functions y = 2 – x and y = 3x – 10 in
the domain 0 ≤ x ≤ 5.
Quadratic graphs
Worked e xa m p le 5
(a) Draw the graph of the function f : x → x2 + x – 6 in the domain – 4 ≤ x ≤ 3.
(b) Estimate, from your graph:
(i) The value of f(2.2)
(ii) The values of x for which f(x) = 0
(iii) The values of x for which x2 + x – 6 = 4
7
Solution
Functions and graphs
(a) This grid will help us to evaluate f(x) for the domain:
x
–4
–3
–2
–1
0
1
2
3
x2
16
9
4
1
0
1
4
9
x
–4
–3
–2
–1
0
1
4
9
–6
–6
–6
–6
–6
–6
–6
–6
–6
y
6
0
–4
–6
–6
–4
0
6
The points are (–4,6), (–3,0), (–2, –4), (–1, –6), (0, –6), (1, –4), (2,0), (3,6)
Here is the graph:
(b)
(i) This means: ‘If x = 2.2, what is the value of y?’ The dotted line from 2.2 to the graph
and then to the y-axis gives us the estimate:
f(2.2) = 1
(ii) This means: ‘If y = 0, find x.’ The graph cuts the x-axis at (–3,0) and (2,0).
Hence the values of x for which f(x) = 0 are –3 and 2.
(iii) This means: ‘If y = 4, find x.’ Draw lines both East and West from 4 on the y-axis.
These give us the estimates:
x = –3.7 and 2.7
Worke d exa m p le 6
(a) Draw the graph of y = 5 + 4x – x2 in the domain –1 ≤ x ≤ 5.
(b) Zero on the x-axis represents midday, while 1, 2, 3, etc. represent 1 p.m., 2 p.m., 3 p.m., etc.
The readings on the y-axis represent the depth of water (in metres) in a harbour. Find:
(i) The depth at 3.30 p.m.
(ii) The greatest depth and the time at which it occurs
Solution
(a)
x
–1
0
1
2
3
4
5
5
5
5
5
5
5
5
5
+4x
–4
0
4
8
12
16
20
–x2
–1
0
–1
–4
–9
–16
–25
y
0
5
8
9
8
5
0
The points are (–1,0), (0,5), (1,8), (2,9), (3,8), (4,5), (5,0)
8
exercise 2
2. Draw the graph of the function:
y = 3x + 1 in the domain –1 ≤ x ≤ 5.
3. (i) Draw the graph of the function:
f : x → 3x – 3 in the domain 0 ≤ x ≤ 5.
(ii) From your graph, estimate the value of
f(3.3).
4. (a) Draw the graph of the function:
f : x → 2x + 3 in the domain –2 ≤ x ≤ 4.
(b) Estimate (from your graph):
(i) The value of f(1.5)
(ii) The value of x for which f(x) = 8
5. (a) Draw the graph of the function:
y = 5 – x in the domain –1 ≤ x ≤ 4.
(b) Estimate from your graph:
(i) The value of y if x = 3.3
(ii) The value of x if y = 5.4
6. (a) Draw the graph of the function:
y = 6 – 2x in the domain –2 ≤ x ≤ 4.
(b) Estimate from your graph:
(i) The value of y if x = 1.3
(ii) The value of x if y = 1.4
7. (a) Draw the graph of the function:
f : x → 4x – 5 in the domain –2 ≤ x ≤ 3.
(b) Estimate from your graph:
Functions and graphs
1. Draw the graph of the function:
f : x → 2x + 1 in the domain –1 ≤ x ≤ 4.
(i) The value of f(0.2)
(ii) The value of x if f(x) = 6
8. (i) Using the same scales and axes, draw the
graphs of the two functions y = 3 – x and
y = x – 1 in the domain 0 ≤ x ≤ 4.
(ii) Use your graphs to solve the simultaneous
equations y = 3 – x and y = x – 1.
9. (i) Using the same scales and axes, draw the
graphs of the two functions y = –2x and
y = 3x – 5 in the domain –1 ≤ x ≤ 3.
(ii) Use your graphs to solve the simultaneous
equations y = –2x and y = 3x – 5.
10. (i) Using the same scales and axes, draw (in the
domain –3 ≤ x ≤ 1) the graphs of the two
functions y = –x and y = 2x + 3.
(ii) Use your graphs to solve the simultaneous
equations y = –x and y = 2x + 3.
11. (i) Using the same scales and axes, draw
the graphs of these two functions in the
domain 0 ≤ x ≤ 5:
f : x → __12 x – 3 and g : x → 3 – x
(ii) Use your graphs to solve the simultaneous
equations y = _12 x – 3 and y = 3 – x.
12. (a) Solve the simultaneous equations y = 2 – x
and y = 3x – 10 using an algebraic method.
(ii) Solve these equations by graphing the
functions y = 2 – x and y = 3x – 10 in
the domain 0 ≤ x ≤ 5.
Quadratic graphs
Worked exa m p le 5
(a) Draw the graph of the function f : x → x2 + x – 6 in the domain – 4 ≤ x ≤ 3.
(b) Estimate, from your graph:
(i) The value of f(2.2)
(ii) The values of x for which f(x) = 0
(iii) The values of x for which x2 + x – 6 = 4
9
Functions and graphs
9. Draw the graph of the function y = x2 + 4x
in the domain –5 ≤ x ≤ 1, x ∈ R.
Use your graph to find:
(i) The value of y when x = 0.3
(ii) The values of x if y = 0
(iii) The values of x for which y = –2
(iv) The least value of y and the value of x at
which it occurs
10. Draw the graph of the function y = x2 – 6 in
the domain –4 ≤ x ≤ 4, x ∈ R.
Use your graph to find:
Find, from your graph:
(i) The values of f(–2.8)
(ii) The values of x for which f(x) = –1
15. Copy and complete the following table for
the function of f : x → 4 – x – 2x2.
x
–3
y
–11
–2
–2
0
1
2
4
25
–11
(i) The value of y when x = 1.4
Draw the graph of y = f(x) in the domain
–3 ≤ x ≤ 2.5, x ∈ R.
(ii) The values of x if y = 0
Use your graph to find:
(iii) The values of x for which y = 5
11. Draw the graph of the function y = x2 – 6x + 5
in the domain 0 ≤ x ≤ 6, x ∈ R.
Take zero on the x-axis to be midnight and
1, 2, 3, etc. to represent 1 a.m., 2 a.m., 3 a.m.,
etc. The readings on the y-axis represent the
temperature (in degrees Celsius).
Find, from your graph:
(i) The temperature at half past five in the
morning.
(ii) The lowest temperature reached and the
time at which it occurred
(iii) The times at which the temperature
was zero
12. Draw the graph of f : x → 2x2 – 3x – 7 in the
domain –2 ≤ x ≤ 3, x ∈ R.
Find, from your graph:
(i) The value of f( 1.7)
(ii) The values of x for which f(x) = 0
13. Draw the graph of the function y = 3x2 – 3x – 4
in the domain –2 ≤ x ≤ 3, x ∈ R.
Use your graph to estimate:
(i) The values of x for which 3x2 – 3x – 4 = 0
(ii) The values of x for which 3x2 – 3x – 4 = 11
10
14. Draw the graph of the function
f : x → 4x2 + 6x – 7 in the domain
–3 ≤ x ≤ 4, x ∈ R.
(i) The value of f(1.7)
(ii) The values of x for which f(x) = 0
(iii) The values of x for which f(x) = 1.5
16. Draw the graph of the function y = x2 – 8x + 16
in the domain 0 ≤ x ≤ 8, x ∈ R.
The readings on the y-axis represent the
speed of a car in metres per second, while
the readings on the x-axis represent the time
(in seconds) as the car approaches a junction.
Use your graph to find:
(i) The speed of the car after 1.3 seconds
(ii) The time that passes before the car stops
(iii) The two times at which the car’s speed is
2 m/s
17. Draw the graph of the function f : x → x2 – x + 4
in the domain –3 ≤ x ≤ 4, x ∈ R.
The graph represents the depth of water
(in metres) in a harbour, from 9 p.m. one
evening until 4 a.m. the next morning (each
unit on the x-axis representing one hour and
zero representing midnight).
Use your graph to estimate:
(i) The depth of water at 3.30 a.m.
(ii) The time when the water was at its
shallowest level
(iii) The two times when the depth was
7.8 m
18. Draw the graph of y = x2 + x – 18 in the
domain 0 ≤ x ≤ 5.
Use your graph to estimate:
(i) The time when the missile reaches
sea-level
(ii) The depth below sea-level after
2.3 seconds
(iii) The height above sea-level after
4.7 seconds
19. Draw the graph of y = 6x – x2 in the domain
0 ≤ x ≤ 6.
y represents the height (in metres) of a ball
thrown straight up into the air from ground
level. x represents the time in seconds after it
is thrown.
Use your graph to find:
(i) The height of the ball after 1.3 seconds
(ii) The greatest height reached
(iii) The time which the ball spends in the air
20. Draw the graph of y = 8 + 2x – x2 in the
domain –2 ≤ x ≤ 4.
y represents the height (in kilometres) of an
aeroplane on a long journey. x represents the
time in hours. Zero represents noon. 1, 2, 3,
etc. represent 1 p.m., 2 p.m., 3 p.m., etc.
Use your graph to find:
(i) The time of take-off
f : x → x2 – x + 1 and g : x → x + 4 in the
domain –2 ≤ x ≤ 4. At what points do the
graphs intersect?
22. On the same piece of graph paper with the
same axes and scales, draw the graphs of the
two functions:
f : x → 2x2 + x – 5 and g : x → 2x + 1 in the
domain –2 ≤ x ≤ 2.
Use the graphs to find the values of x for
which:
(i) f(x) = 0
(ii) f(x) = g(x)
Functions and graphs
y represents the height in metres of a missile
above (and below) sea-level. The missile is
launched from a submarine and rises out of
the sea into the air. x represents the time (in
seconds) after the launch.
21. On the same piece of graph paper with the
same axes and scales, draw the graphs of the
two functions:
23. On the same piece of graph paper with the
same axes and scales, draw the graphs of the
two functions:
f : x → x2 – 3x + 3 and g : x → 3 – x in the
domain –1 ≤ x≤ 4.
Use the graphs to find the value(s) of x for
which:
(i) f(x) = 6
(ii) g(x) = 0
(iii) f(x) = g(x)
24. On the same piece of graph paper with the
same axes and scales, draw the graphs of the
two functions:
f : x → 4x – x2 and g : x → x2 – 8x + 16 in the
domain 0 ≤ x ≤ 6.
Use the graphs to find the values of x for
which:
(i) f(x) = 0
(ii) g(x) = 0
(iii) f(x) = g(x)
(ii) The length of time spent in the air
(iii) The greatest height reached and the time
at which this occurred
11
Answers
Exercise 1
Exercise 3
1. 17 2. 13 3. 73 4. 20 6. a = 17, b = –11, c = 4
7. p = 5, q = 0, r = –2, s = 8 8. {3, 8, 13} 9. (i) 4
(ii) {4, 3, 2, 1, 0, –1} 10. {1, 2, 5} 11. {14, 17, 20}
12. {–10, –3, 4, 11} 13. {4, 29} 14. (i) 18 (ii) 50
(iii) 200 15. {7, 22, 43, 70} 16. {7, 37, 87}
17. (i) 27 (ii) –43 (iii) 5 (iv) 2 18. (i) 5, 11, 21
(ii) No (iii) 5 19. 19, –86, 10 20. 21, 3, –3
21. (i) {1, 3, 5} (ii) {0, 1, 2, 3, 4, 5} 22. (i) {2, 3}
(ii) {0, 1, 2, 3} 23. 38 24. (i) 50 (ii) 2 (iii) 92
25. 0 and 0 26. (i) 18 (ii) 0 (iii) 0 27. (i) 10 (ii) 0
(iii) 0 28. (i) 2 (ii) 1½ 29. 10 30. (i) 3 (ii) x ≤ 3
1. (i) 3.3 (ii) 1,3 2. (i) 4.3 (ii) 1.6,4.4 3. (i) 2.75
(ii) −1,4 (iii) −0.6,3.6 4. (i) −8.25 (ii) −4,3
(iii) −3.4,2.4 5. (i) −3.6 (ii) −1.8,2.8 (iii) −2.5,3.5
6. (i) 4.25 (ii) −0.4,2.4 (iii) −2 7. (i) −1.25
(ii) −1,4 (iii) 0.4,2.6 8. (i) −2.75 (ii) −3.2,1.2
(iii) −5,−1 9. (i) 1.3 (ii) −4,0 (iii) −3.4,−0.6
(iv) −4,−2 10. (i) −4 (ii) −2.45,+2.45
(iii) −3.3,+3.3 11. (i) 2.25° (ii) −4° at 3 a.m.
(iii) 1 a.m. and 5 a.m. 12. (i) −6 (ii) −1.3,2.8
13. (i) −0.7,1.7 (ii) −1.8,2.8 14. (i) 7.5
(ii) −2.2,0.6 15. (i) −3.5 (ii) −1.7,1.2
(iii) x = −1.4,0.9 16. (i) 7.3 m/s (ii) 4 s
(iii) 2.6 and 5.4 seconds 17. (i) 12.75 (ii) 00.30 a.m.
(iii) 10.30 p.m. and 2.30 a.m. 18. (i) 3.8 (ii) 10.4 metres
(iii) 8.8 metres 19. (i) 6.1 m (ii) 9 m (iii) 6 seconds
20. (i) 10 a.m. (ii) 6 hours (iii) 9 km; 1 p.m.
21. (−1,3) and (3,7) 22. (i) −1.85,1.35 (ii) −1.5,2
23. (i) −0.8,3.8 (ii) 3 (iii) 0,2 24. (i) 0,4
(ii) 4 (iii) 2,4
Exercise 2
3. (ii) 6.9 4. (b)(i) 6 (ii) 2.5 5. (b)(i) 1.7 (ii) –0.4
6. (b)(i) 3.4 (ii) 2.3 7. (b)(i) –4.2 (ii) 2.75 8. (ii) (2,1)
9. (ii) (1,−2) 10. (ii) (−1,1) 11. (ii) (4,−1) 12. (3,−1)
13. (c)(i) 250 mins (ii) 3.75 kg 14. (c)(i) 63° (ii) 28°
15. (c)(i) 26 (ii) 18
12