Educerie · SAT · Math
Algebra · ALG.3 Linear functions
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Read function notation | If f(x) = 4x − 9, what is f(3)? |
| Say what the slope means in context | What is the best interpretation of 4.5 in this context? |
| Say what the intercept means in context | Almost always the starting value |
| Build a linear model from a description | Two quantities, a rate, a starting point |
| Work backwards from an output | For what value of x does f(x) = 31? |
1The idea in one paragraph
A linear function is f(x) = mx + b, and the test cares less about solving it than about what m and b mean when x is hours and f(x) is lira. The coefficient of x is a rate — something per something. The constant is the value when x is zero. More questions on this section turn on that single sentence than on any calculation.
In h(t) = 18 + 4.5t, the balloon starts at 18 metres and rises 4.5 metres each minute. Say it in words with the units in place, and interpretation questions answer themselves.
2Function notation is a set of instructions
f(x) = 4x − 9 says: take the input, multiply by 4, subtract 9.
The letter is decoration; the machine is what matters.
Nested notation is read from the inside out: f(g(2)) means work out g(2) first, then put that number into f.
3Interpretation questions, and the trap in them
The cost C, in lira, of printing n copies is C(n) = 250 + 1.4n.
- 1.4 is the cost of each additional copy — a rate, per copy.
- 250 is the cost when no copies are printed — a fixed set-up cost.
The wrong options swap these two, or turn the rate into a total: "the printer charges 1.4 lira in total" or "the set-up costs 250 lira per copy". Read the units in the question — in lira, per copy — because the units tell you which number is which.
4Building a model from a sentence
Three things to extract, in this order: what is being counted (the input), what is being measured (the output), and the rate between them.
A rate that reduces the quantity is negative. That is the most common single wrong answer in this unit.
5From a table
| t | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| f(t) | 12 | 19 | 26 | 33 |
If the table has no t = 0 row, use point-slope instead of guessing.
6Domain and range in context
A model built from a real situation only makes sense over part of the number line. A tank that empties in 60 minutes has a model that holds for 0 ≤ t ≤ 60; asking for V(200) is asking about a tank that has been empty for two hours. Questions phrased "for which values of t is the model valid" are asking exactly this.
Where points are lost
- Swapping rate and starting value in an interpretation question.
- Missing the units. Per hour, in lira, each — they name the rate.
- Taking the rate as a total.
- Getting the sign wrong on a decreasing quantity.
- Solving f(x) = 0 when the question asked for f(0), or the reverse.
- Working nested notation outside-in.
- Ignoring the context's limits on sensible inputs.
Work it right
- Name the input and the output, with units.
- Identify the rate: the number attached to the variable.
- Identify the starting value: the number standing alone.
- Say both in a sentence with units before looking at the options.
- Check the sign against whether the quantity grows or shrinks.
Try it
Q1. The function C(n) = 180 + 2.5n gives the cost, in lira, of a print run of n posters. What is the best interpretation of 2.5 in this context?
A) Each poster costs 2.5 lira to print. B) The print run costs 2.5 lira in total. C) The set-up charge is 2.5 lira. D) 2.5 posters can be printed for one lira.
Q2. If f(x) = 5x − 8, what is the value of x when f(x) = 27?
A) 3.8 B) 7 C) 19 D) 127
Q3. A reservoir holds 8,600 m³ and loses 120 m³ a day to evaporation during a drought. Which function gives the volume V, in m³, after d days?
A) V(d) = 120d − 8600 B) V(d) = 8600 + 120d C) V(d) = 8600 − 120d D) V(d) = 120 − 8600d
Q4. The table shows a linear function g.
| x | 1 | 3 | 5 | 7 |
|---|---|---|---|---|
| g(x) | 9 | 17 | 25 | 33 |
What is g(0)? (Type your answer.)
Q5. If f(x) = 2x + 1 and g(x) = x² − 3, what is f(g(3))?
A) 6 B) 11 C) 13 D) 49
In one breath
The number attached to the variable is a rate — something per something — and the number standing alone is the value when the variable is zero. Say both out loud with their units before you look at the options, because the wrong answers are built by swapping them, by turning the rate into a total, or by making a falling quantity rise. Function notation is just a machine: f(3) means feed in 3, and f(x) = 31 means run it backwards.
Answers
Q1. A. 2.5 is attached to n, so it is cost per poster B turns a rate into a total. C gives 2.5 the job of the constant, which is 180. D inverts the rate, reading lira per poster as posters per lira.
Q2. B — 7. run the machine backwards
A subtracts the 8 instead of adding it, then divides: 19 ÷ 5. C is 27 − 8, the same slip with no division. D computes f(27) instead of solving f(x) = 27.
Q3. C. starts full, falls by 120 each day B makes the reservoir fill during a drought, and is the sign error this question exists to catch. A and D attach the numbers to the wrong roles entirely.
Q4. 5. find the slope, then step back to x = 0
Q5. C — 13. inside out
B computes f(3) then g of it wrongly; A stops at g(3); D squares the result of f(3).
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Algebra, skill/knowledge testing point "Linear functions"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
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