PSAT Advanced Math — function notation and what it stands for

Function notation is a piece of shorthand that costs marks until it is read fluently, and then costs none. The whole of it is this: $f(x)$ names the output of the rule $f$ when its input is $x$, and it never means $f$ multiplied by $x$. Almost everything in this material follows from taking that sentence seriously.

Twelve questions, in five groups. Three are evaluation, and one of them runs the other way: given that the output is seventeen, find the input. That direction is worth practicing separately, because the rule has to be undone rather than applied, and the wrong options are the middle line of the working mistaken for the answer and the operations applied in the wrong order. A third evaluates at a negative input, where the brackets are the whole question — four minus three times minus two is ten, not minus two.

Two are compositions. The inner function is worked out first and its output becomes the input of the outer one, which sounds obvious and is reversed by a good share of candidates under time. In both questions the options include the inner result on its own, the outer function applied to the original input, and the two functions composed the other way round, so choosing carelessly lands on a number that exists rather than on nothing.

Three read a function that has been defined in words with units attached: a delivery cost in dollars as a function of miles, energy in kilowatt-hours as a function of hours of sunlight, pages as a function of minutes. The question is what a statement like $C(10)=25$ says, and the reliable wrong answer swaps the input with the output — twenty-five miles for ten dollars rather than ten miles for twenty-five.

Two give a short list of input-output pairs in words and ask for the next value or for the rule behind them, where the trap is a rule that fits the first pair and fails the second. Testing every pair takes ten seconds and settles it.

The last two are about domain: a baker limited to ten batches, a tank holding fifteen gallons. The arithmetic would accept any number at all, and the situation will not, which is the distinction being examined.

Every situation here was invented for the exercise, and nothing is reproduced from any published test.

• Read $f(x)$ as the output of a rule at a given input, and never as a multiplication
• Evaluate a function at a stated input, including a negative one, keeping the brackets
• Work backwards from a stated output to the input that produced it
• Compose two functions from the inside out, and recognize the value the opposite order would give
• Say what a statement such as $C(10) = 25$ means in the situation the function describes, with the right units on each side
• Find a rule from a list of pairs by testing every pair rather than the first
• Narrow a domain from what the arithmetic allows to what the situation permits

Every rule and situation here was written for this exercise; there are no graphs, and any list of values is stated in words inside its question. Among them: $f(x) = 2x + 5$ solved backwards from an output of seventeen; $h(x) = 5 - 2x$ at $x = 4$; $f(x) = 4 - 3x$ at $x = -2$; the pairs $f(x) = x + 3$ with $g(x) = 2x$, and $f(x) = 3x$ with $g(x) = x - 1$, composed in both directions; a delivery cost $C(m) = 2m + 5$ where $C(10) = 25$; energy $E(h) = 4h$ where $E(5) = 20$; pages $P(t) = 15t$; the values 3, 5, 7 and 9 continued; the pairs 1, 4, 7 and 10 fitted by $g(x) = 3x + 1$; a baker limited to ten batches; and a tank holding fifteen gallons. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not an exam centre.