Linear functions

This skill point covers three things that look different and are the same thing: reading function notation (evaluating f at a given input, or solving for the input that produces a given output), interpreting what a number or a statement like f(12) = 340 means inside a described situation, and building the function yourself from a paragraph of English. The arithmetic in all three is one multiplication and one addition. What is being tested is whether the quantities keep their meaning while you move them.

College Board publishes the domain weight: Algebra is about 35% of the Math section, the largest of the eight domains. What it does not publish is how that 35% divides among the five Algebra skill points, so any per-skill count you see quoted — "expect three or four linear-function items" — is a prep-industry estimate reverse-engineered from released material, not an official figure. Treat it as a planning assumption rather than a fact.

What distinguishes this skill from its neighbour, linear equations in two variables: that one asks you to relate x and y. This one names the relationship, gives it a letter, and then asks questions about the naming. That extra layer is where the points move. A student who can solve 3x − 12 = −6 flawlessly can still lose the item by not noticing that the question asked for f(−6) instead — same rule, same number, different answer.

If you can look at f(x) = 3x − 12 and produce f(−6) without hesitating, skip ahead; nothing here will be new. If the notation itself is the thing slowing you down, this is the highest-value block on the page, because f(x) is not confined to this skill — it is the language the rest of the Math section is written in.

A function is a rule that takes one input and returns exactly one output. That is the entire definition. f is the rule's name. x is a placeholder for whatever goes in. f(x) is what comes out. So f(x) = 3x − 12 is the rule spelled out: take the input, multiply by 3, subtract 12, hand back the result.

Fix this first, before anything else: f(x) is not f times x. The parentheses here are not multiplication — they are holding the input. Reading them as multiplication is the error that makes the whole notation feel arbitrary, and it does not recover on its own.

f(−6) is an instruction, not an expression to simplify. It says: everywhere the rule has an x, write −6 instead. f(−6) = 3(−6) − 12 = −18 − 12 = −30. Write the input inside its own parentheses when you substitute. That single habit prevents most of the sign errors in this skill.

Now the distinction that decides more items than any other on this page. The number can appear in two places, and its position tells you which job you have. f(−6) = ? puts the number INSIDE the parentheses: the input is given, find the output — substitute and compute. f(x) = −6 puts the number on the OTHER SIDE of the equals sign: the output is given, find the input — set the rule equal to −6 and solve. Same function, same number, two different questions, and two different answers: −30 and 2.

Notation, translated once and for all: f(a) = b means "when the input is a, the output is b," and it also means "the point (a, b) is on the graph of y = f(x)." Those are the same statement said twice. y and f(x) are interchangeable — writing y = f(x) just says the vertical axis holds the outputs.

Two consequences the test uses constantly. f(0) is the y-intercept: the output when the input is zero. A zero of f — also called a root, and visible on a graph as the x-intercept — is an input that makes the output zero, which you get by solving f(x) = 0. They sound alike, they are opposite ends of the same line, and questions phrase them in language designed to be confusable.

The letters mean nothing mathematically and everything practically. h(t), C(m), P(x), V(d) all behave identically; the exam picks letters that remind you what the quantities are — h for height, t for time, C for cost, m for minutes. When an item hands you an unfamiliar letter pair, read the sentence that defines them before touching the algebra. That is not the slow way through the item; skipping it is what makes the item slow.

Every linear model is m and b in disguise: f(x) = (rate)(x) + (starting value). The work is not writing that down — it is deciding, from a paragraph of English, which number is which and what units each one carries. Run four steps in this order, every time, and the decision stops being a judgement call.

Step 1 — name the input and the output, with units, in writing. "Input: t, hours since the pump started. Output: w, liters remaining." Ten seconds. It is the step everyone skips, and it is the step that prevents every unit error below.

Step 2 — find the rate. It is attached to the words per, each, every, for every, or a phrase like "an additional." Then do the part that actually matters: convert it into output-units per ONE input-unit. "$54 for every 4 posters" is not a rate you can use; $13.50 per poster is. "90 liters every 4 minutes," with the input measured in hours, is not a rate you can use; 1,350 liters per hour is. Finally attach a sign: charging, gaining, rising, growing are positive; draining, descending, depreciating, losing are negative.

Step 3 — find the starting value: the output when the input is exactly zero. In a cost problem it is what is owed before a single unit is bought; in a physical problem it is the measurement at the moment the clock starts. Read the wording carefully, because "at the start" and "at input zero" are not always the same moment — "after the first hour the tank held 400 liters" is a data point, not the intercept.

Step 4 — write f(x) = (signed rate)(x) + (starting value), then test it twice. Once at x = 0, which must return the starting value. Once at any other value the passage states, which must return what the passage says it returns. That second check costs five seconds and catches every sign error and every unit error you are capable of making.

When the description hands you two data points instead of a rate and a start, the same machine runs with one extra step in front of it. Translate the notation into points — f(3) = 41 and f(8) = 26 are the points (3, 41) and (8, 26) — then compute the slope as (26 − 41)/(8 − 3) = −15/5 = −3, then substitute either point back to find the constant. Keep the point order identical in the numerator and the denominator; flipping it in only one of the two places is the standard sign error here.

At this point the notation is not what separates 1500 from 1600. Six things are: composed and transformed functions, change over an interval rather than per unit, contextual domain limits, descriptions that sound linear and are not, items holding two functions at once, and a short list of execution errors that survive knowing everything above.

Transformations, stated so you never have to re-derive them under time pressure. Start from f(x) = mx + b. g(x) = f(x) + k adds k to every output: same slope, intercept b + k. g(x) = f(x + k) substitutes x + k for x, giving m(x + k) + b = mx + (mk + b): same slope, intercept moved by m times k — and note that a PLUS inside the parentheses moves the graph LEFT, which is the opposite of what it reads like. g(x) = k·f(x) multiplies everything: slope km, intercept kb. g(x) = f(kx) multiplies only the slope: slope km, intercept unchanged. The two that look alike, f(x + k) and f(kx), do entirely different things, and an item using one of them will offer the other's answer as a choice.

Change over an interval is the slope times the interval's length — not the slope, and not a level. If E(t) = 15,400 + 260t, then enrollment rises by 260 in one year, by 1,300 over any five-year stretch, and E(5) = 16,700 is where enrollment stands, not how much it moved. Three different numbers, all correct answers to different questions, and a well-built item puts all three in front of you. The reverse direction appears too: "the model predicts a fall of 900 over the period, so what is the yearly rate" is a division, not a reading.

Contextual domain. A model built from a described situation is valid only over a window, and the exam knows it. A 4,500-liter tank draining at 1,350 liters per hour has a model that returns negative liters after about 3.3 hours, so a question asking what remains after 5 hours is either testing whether you notice or has restricted the domain in the setup — read the restriction if one is printed. Watch too for answers the situation forces to be whole: people, buses, complete deliveries. When the algebra returns 6.4 and the context counts buses, the answer is 7 or 6 depending on which way the sentence rounds, and the sentence always says which.

Not every description is linear, and the test writes descriptions that sound linear and are not. "Decreases by 12% each year" is a constant multiplier, so it is exponential. "Doubles every six days" is exponential. "Falls by 40 units per year" is linear. The tell is amount versus percentage: a linear model changes by the same NUMBER every step, so if you cannot name that number, the relationship is not linear and nothing on this page applies to it.

Two functions in one item. "For what value of x is f(x) = g(x)" asks for the input at which the two rules agree: set the expressions equal and solve for x. Then reread the last line, because roughly half the time the item wants the common OUTPUT and the x you just found is sitting there as a distractor. Same shape in context: "after how many hours do the two tanks hold the same amount" wants the time; "how much does each tank hold when they are equal" wants the volume, which takes one more substitution.

One graph-reading variant that catches strong students: a linear function shown on axes with different scales. Slope is the change in the output axis's units divided by the change in the input axis's units — never the visual steepness. A line that looks like it rises at forty-five degrees has slope 1 only when one grid square means the same amount on both axes, which on this exam it frequently does not.

Execution errors, which is where the remaining points actually live. Substituting a negative input without parentheses. Reporting b when the question named a different input. Rounding a rate before the final step — 32/80 is exactly 0.4, but a rate carried as 0.33 instead of 1/3 will miss a student-produced response by enough to be marked wrong. Answering with t when the passage counts years after 1990 and the question asks for the year, or with the year when it asks for t. And the format one worth knowing cold: student-produced responses do not accept mixed numbers, so an answer of two and a half is entered as 5/2 or as 2.5 and never as 2 1/2, which the scoring reads as twenty-one halves.