What Is a Function? — Print
A function is a rule with no ambiguity: one input, exactly one output. That is the whole definition.
What the practice 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.
Press B4 on a vending machine and a packet of crisps drops out. Press B4 again tomorrow — crisps again.
That reliability is the whole point of the machine. A machine that gave crisps sometimes and a drink other times would be useless, and nobody would put money in it.
Notice what is allowed, though. B4 and C2 might both give crisps. Two buttons, one snack — that is fine.
What is never allowed is one button giving two different things.
A function is exactly that machine: each input gives exactly one output. Repeats on the output side are fine; a repeated input with different outputs breaks it.
The test for a function
Ask one question of every input: does it have more than one output? If yes, not a function. If no, it is one.
The input 2 may not go to both 5 and 9. But two different inputs may share an output — that is perfectly allowed.
A vending machine is the image to hold: press B4 and you get the same thing every time.
Repeats are fine on the output side
{(1, 4), (2, 4), (3, 4)} is a function. {(1, 4), (1, 9)} is not. Look for a repeated input, nothing else.
The vertical line test
On a graph, slide an imaginary vertical line across. If it ever touches the curve twice at once, that x has two y values, so it is not a function.
Straight lines pass, except vertical ones. Parabolas pass. Circles fail.
It is the same test as before, drawn instead of listed.
Use the edge of a ruler
Lay a ruler vertically on the graph and slide it. Two crossings anywhere means not a function. Physically doing it is faster than reasoning about it.
Function notation
f(x) is a name for the output, not multiplication. f(x) = 3x + 1 and y = 3x + 1 say the same thing.
f(4) means substitute 4 for x: f(4) = 3(4) + 1 = 13.
Going the other way, "f(x) = 13, find x" means solve 3x + 1 = 13, giving x = 4.
f(4) is not f × 4
The brackets mean "evaluate at". If f(x) = x2 then f(3) = 9, not 3f.
Linear or not
A function is linear if its graph is a straight line — equivalently, if equal steps in x give equal steps in y.
In a table, take the differences between consecutive y values. Constant differences mean linear.
y = x2 gives differences 1, 3, 5, 7 — not constant, so not linear. y = 3x gives 3, 3, 3 — linear.
Check the x steps too
Constant y differences only prove linearity if the x values also step evenly. Check both columns.
Describing a graph in words
Say where it increases, where it decreases, and where it is flat. Say whether it is linear or curved.
Name the intercepts if they matter, and any maximum or minimum.
These qualitative descriptions are worth real marks and need no calculation at all.
Worked examples
Is it a function?
Is {(2, 5), (4, 9), (2, 7), (6, 1)} a function?
- List the inputs: 2, 4, 2, 6 Only the inputs matter for this test.
- The input 2 appears twice, going to 5 and to 7. One input, two different outputs.
- So it is not a function. The definition is broken by that one pair.
No — the input 2 has two outputs
Evaluating
If f(x) = 2x2 - 5, find f(-3).
- Replace every x with -3: f(-3) = 2(-3)2 - 5 Substitute with brackets around the negative.
- (-3)2 = 9 Square first: a negative squared is positive.
- 2(9) - 5 = 18 - 5 = 13 Then multiply, then subtract.
f(-3) = 13
Linear from a table
Is this linear? x: 1, 2, 3, 4 and y: 5, 8, 11, 14.
- x steps: +1, +1, +1 — even. Check the inputs first.
- y differences: 3, 3, 3 8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3.
- Constant, so linear with slope 3. At x = 1, y = 5, so b = 2: the rule is y = 3x + 2.
Yes — linear, y = 3x + 2
Words to know
- Function
- A relationship where each input gives exactly one output.
- Input / output
- What goes in (x) and what comes out (y).
- Domain
- The set of allowed inputs.
- Range
- The set of outputs that actually occur.
- Function notation
- f(3) = 7 means: feed in 3, get out 7.
Mistakes to watch for
Rejecting a function because an output repeats. {(1, 4), (2, 4), (3, 4)} is a perfectly good function — three buttons, same snack. Only a repeated input with different outputs disqualifies it.
Reading f(x) as f times x. The brackets mean "evaluate at," so f(3) is the output when the input is 3. If f(x) = x2 then f(3) = 9, not 3f.
Forgetting brackets around a negative. f(-3) with f(x) = x2 means (-3)2 = 9. Typing -32 gives -9, because then the exponent only attaches to the 3.
Calling a table linear from the y column alone. Constant differences only mean linear if the x values also step evenly. Check both columns, or a table that jumps 1, 2, 4 in x will fool you.
The notation f(x) was introduced by Leonhard Euler in 1734. He also gave us e, π as we write it, i, and the summation sign Sigma.
Euler went blind in his sixties and still produced roughly half of his total work afterwards, dictating it to his sons.
He is often called the most prolific mathematician in history; it took a society of editors more than a century just to publish his collected papers.
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