A function gives one output per input. One. That is the whole test.
Which is a function?
Which set of ordered pairs is a function?
In C every x-value — 2, 5 and 7 — appears only once, so each input has exactly one output. Repeating the output 9 is perfectly allowed. A, B and D each repeat an x-value with two different partners, which is what breaks the rule. Check the x's for repeats; the y's do not matter.
{(1, 4), (2, 7), (3, 10)}
a relation
domain = {1, 2, 3}
all the x-values
range = {4, 7, 10}
all the y-values
Both are written as sets, so a repeated value is listed only once. The domain of {(1, 5), (1, 8)} is just {1}.
One input, one output. Outputs may repeat as much as they like.
Three different people can have the same height. What cannot happen is one person having two heights. That is the asymmetry the rule captures.
The test is the same rule drawn: if any vertical line meets the graph more than once, that x has two outputs.
f(3) means "put 3 in", not "f times 3".
f(x) = 2x² − 5
find f(−3)
f(−3) = 2(−3)² − 5
brackets around the input
= 2(9) − 5 = 13
the output
The letter is only a label. f(x), g(x) and h(x) let you talk about several functions at once without confusing their outputs.
f(x) = 3x + 7, and f(x) = 22
what went in?
3x + 7 = 22
an ordinary equation
x = 5
check: 3(5) + 7 = 22 ✓
f(g(x)) means: run g first, then feed its output into f.
If f(x) = 2x − 1 and g(x) = x², find f(g(3)).
g(3) = 3² = 9
the inner machine first
f(g(3)) = f(9)
its output becomes the new input
f(9) = 2(9) − 1 = 17
the answer
Starting with f gives f(3) = 5, and then g(5) = 25. That is g(f(3)), a different question with a different answer. The function written closest to the number goes first.
The same rules, in the three places students hesitate.
A rule with a fraction: f(x) = 2x + 1x − 3. Find f(1) and f(3).
f(1) = 2(1) + 11 − 3 = 3−2
top and bottom separately
f(1) = −32
a negative bottom makes the answer negative
f(3) = 70
undefined: 3 is not in the domain
An output need not be a whole number, and some inputs have no output at all.
Backwards with a square: f(x) = x² + 1 and f(a) = 26. Find a.
a² + 1 = 26
set the rule equal to the output
a² = 25
subtract 1
a = 5 or a = −5
check: (−5)² + 1 = 26 ✓
Two inputs can share one output. Still a function — and both answers count.
Is {(3, 1), (4, 2), (3, 1)} a function?
x = 3 appears twice
look at its partners
both times the partner is 1
the same pair written twice
yes, a function
3 still has only one output
A repeated x only breaks the rule when its partners are different.
Tests ask this three ways: pick the set or graph that is (or is not) a function, evaluate f at a negative number or inside another function, or give the output and ask for the input. For sets, scan only the first coordinates for a repeat with a different partner; for f(a) = k, write the equation at once. The 10-second check is to put your answer back into the rule and see the given output come out.
Solve for the input
If g(x) = x² − 4x, for which value of x does g(x) = 0 besides x = 0?
Set the output to zero: x² − 4x = 0, so x(x − 4) = 0, giving x = 0 or x = 4. Check: g(4) = 16 − 16 = 0 ✓ C gives g(−4) = 16 + 16 = 32, not 0. Notice this is a function even though two different inputs share the output 0 — repeated outputs are allowed.
Hint: "g(x) = 0" means set the expression equal to zero and solve. Factor rather than dividing by x, or you will lose a solution.
