What Is a Function? — Practice
A function is a rule with no ambiguity: one input, exactly one output. That is the whole definition.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Function — A rule pairing each input with exactly one output.
- Slope — How much y changes for each 1 that x increases.
- Domain — The set of allowed inputs.
- Range — The set of outputs that actually occur.
- Parabola — The U-shaped graph of a quadratic function.
Problem 1
Is {(1, 3), (2, 5), (3, 7)} a function?
- Yes
- No, because the outputs differ
- No, because 1 is an input
- Not enough information
Worked answer
Yes
Inputs are 1, 2, 3 — all different
Each has exactly one output
So yes
Problem 2
Is {(4, 1), (5, 2), (4, 9)} a function?
- No, because the input 4 has two outputs
- Yes
- No, because 9 is too large
- Yes, because all outputs differ
Worked answer
No, because the input 4 has two outputs
4 appears twice, with 1 and with 9
One input must have exactly one output
So it is not a function
Problem 3
If f(x) = 3x + 2, find f(5).
Worked answer
17
3(5) + 2
= 15 + 2 = 17
Problem 4
If g(x) = x2 - 4, find g(3).
Worked answer
5
32 = 9
9 - 4 = 5
Problem 5
If f(x) = 2x - 7, find x when f(x) = 9.
Worked answer
8
2x - 7 = 9
2x = 16
x = 8
Problem 6
If h(x) = -x2 + 3x, find h(-2).
Worked answer
-10
-(-2)2 + 3(-2)
= -4 - 6
= -10
Problem 7
Which graph fails the vertical line test?
- A circle
- A straight line with slope 2
- A parabola opening up
- A horizontal line
Worked answer
A circle
A circle is crossed twice by most vertical lines
So one x gives two y values
It is not a function
Problem 8
Is this table linear? x: 1, 2, 3, 4; y: 2, 6, 18, 54. Answer yes or no.
Worked answer
no
Differences: 4, 12, 36 — not constant
Ratios: 3, 3, 3 — constant
So it is exponential, not linear
Problem 9
A table has x: 0, 2, 4, 6 and y: 5, 11, 17, 23. Write the rule.
Worked answer
y = 3x + 5
y steps by 6 each time x steps by 2
m = 3
At x = 0, y = 5, so y = 3x + 5
Problem 10
For f(x) = 1x - 4, which value must be excluded from the domain?
Worked answer
4
x - 4 = 0 when x = 4
That input has no output
Exclude x = 4
Problem 11
If f(x) = 5 - 2x, find f(0) + f(4).
Worked answer
2
f(0) = 5
f(4) = 5 - 8 = -3
5 + (-3) = 2
Problem 12
Which set is a function?
- {(1, 5), (2, 5), (3, 5)}
- {(5, 1), (5, 2), (5, 3)}
- {(1, 2), (1, 3)}
- {(0, 0), (0, 1)}
Worked answer
{(1, 5), (2, 5), (3, 5)}
The first has inputs 1, 2, 3 — all different
The others all repeat an input
Problem 13
If f(x) = x2 + 1 and g(x) = 2x - 3, find f(g(4)).
Worked answer
26
g(4) = 8 - 3 = 5
f(5) = 25 + 1 = 26
Problem 14
A function has f(2) = 7 and f(6) = 19 and is linear. Find f(10).
Worked answer
31
m = 19 - 76 - 2 = 3
From f(6) = 19, four more steps of 3
19 + 12 = 31
Problem 15
For f(x) = x2 with domain {-2, -1, 0, 1, 2}, list the range.
Worked answer
{0, 1, 4}
4, 1, 0, 1, 4
Distinct values: 0, 1, 4
Problem 16
If f(x) = 3x + k and f(4) = 20, find k.
Worked answer
8
3(4) + k = 20
12 + k = 20
k = 8