Functions — Practice
A function is a machine: one input goes in, exactly one output comes out. The rule is the machine.
What these questions cover
These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Slope — How much y changes for each 1 that x increases.
- y-intercept — The output when x = 0 — where the line crosses the vertical axis.
- Linear function — One whose graph is a straight line: y = mx + b.
Problem 1
If f(x) = 3x + 2, find f(4).
Worked answer
14
3(4) + 2 = 14
Problem 2
A table gives x = 0, 1, 2, 3 and y = 7, 11, 15, 19. What is the slope?
Worked answer
4
11 - 7 = 4, and the jump stays 4
Slope = 4
Problem 3
What is the y-intercept of y = 5x - 3?
Worked answer
-3
At x = 0: y = -3
Problem 4
If f(x) = 2x - 7, find f(5).
Worked answer
3
2(5) - 7 = 3
Problem 5
What is the slope of y = 4x + 1?
Worked answer
4
Slope = 4
Problem 6
A table gives x = 0, 1, 2, 3 and y = 7, 11, 15, 19. Find y when x = 10.
Worked answer
47
y = 4x + 7
4(10) + 7 = 47
Problem 7
A line passes through (2, 9) and (5, 21). Find its slope.
Worked answer
4
21 - 95 - 2 = 123 = 4
Problem 8
A line passes through (2, 9) and (5, 21). Find its y-intercept.
Worked answer
1
9 = 4(2) + b
9 = 8 + b, so b = 1
Check with (5,21): 20 + 1 = 21 ✓
Problem 9
A gym charges $30 to join plus $22 a month. What does a year cost?
Worked answer
294
22 × 12 = 264
264 + 30 = 294
Problem 10
Is y = x2 a linear function?
- Yes
- No
Worked answer
No
x = 1, 2, 3 gives y = 1, 4, 9
Jumps are +3 then +5 — not constant
So it is not linear.
Problem 11
If f(x) = 3x + 2, find x when f(x) = 20.
Worked answer
6
3x = 18
x = 6
Problem 12
Find the slope of the line through (1, 5) and (4, 14).
Worked answer
3
14 - 54 - 1 = 93 = 3
Problem 13
A table gives x = 1, 2, 3 and y = 5, 8, 11. Write the rule in the form mx+b.
Worked answer
3x+2
Jump is 3, so slope = 3
At x = 1, y = 5, so 3(1) + b = 5 and b = 2
y = 3x + 2
Problem 14
A line has slope -2 and passes through (3, 1). Find its y-intercept.
Worked answer
7
1 = -2(3) + b
1 = -6 + b
b = 7
Problem 15
Which table shows a linear function?
- y = 2, 4, 8, 16
- y = 3, 7, 11, 15
- y = 1, 4, 9, 16
- y = 5, 6, 8, 11
Worked answer
y = 3, 7, 11, 15
3, 7, 11, 15 jumps by 4 every time — linear
The others double, or jump by changing amounts.
Problem 16
A tank holds 50 L and drains at 4 L per minute. After how many minutes is it empty?
Worked answer
12.5
50 - 4x = 0
4x = 50
x = 12.5 minutes