ACT® · Math: Functions — Properties of Functions
Interpreting Function Notation in Context (F 507)
Function notation f(x) = y is a compressed sentence: it says "when the input is x, the output is y." To unpack it, you need to know what the input counts and what the output measures.
A function written as C(12) = 47 looks like a single piece of math, but it's really a tiny story with two facts glued together: an input and the output it produces. If C(m) is the cost in dollars of a taxi ride of m miles, then C(12) = 47 says a 12-mile ride cost 47 dollars. The number inside the parentheses is always the input, and the number the whole expression equals is always the output. Read it out loud, swapping in the units: that habit turns an abstract equation into a sentence you can check for sense.
Going the other direction, from words to notation, means spotting which quantity is doing the counting (the input) and which quantity depends on it (the output). 'Twelve miles costs forty-seven dollars' becomes C(12) = 47 once you decide C depends on miles. For a town's population, P(t) = t years after 2020, so P(5) = 18000 means 5 years after 2020, the population was 18000 people — not that the population is 18000 at year zero, and not that t itself equals 18000.
This is also where a classic trap lives: h(0) is tempting to read as 'height is zero,' but 0 here is an input, the starting time, not an output. A drone that lifts off 30 feet up and climbs 5 feet per second has h(0) = 30, not 0. Once you're comfortable reading notation this way, the same skill carries into domain and range and into the four representations, since a table, graph, or equation is just another costume for the same input-output story.
Key terms
- Input
- The value placed inside the parentheses, representing what the function counts, like miles or years.
- Output
- The value the whole function expression equals, representing the resulting quantity, like dollars or population.
- Function notation
- A shorthand like f(x) = y that pairs one input x with exactly one output y.
- Context sentence
- The plain-English translation of a function statement, built by naming the input's and output's units.
Watch it explained
- How it worksHow do you translate the sentence 'twelve miles costs forty-seven dollars' into function notation?
- Worked exampleWalk me through what P(5) = 18000 means when P(t) is a town's population t years after 2020.
- Common mistakeWhy doesn't h(0) always mean the height is zero?
- Going furtherWhat changes about reading f(t) when t is counted in hours instead of years?
Check yourself
End of topic.