SAT Linear Functions: Practice Questions & Study Guide
Understanding linear functions as rules mapping inputs to outputs, evaluating them, and interpreting function notation in real-world models.
Understanding Linear Functions
A linear function is a rule of the form f(x) = mx + b that assigns exactly one output to each input. The key conceptual shift from linear equations to linear functions is the focus on the relationship as a whole rather than just a single solution. On the test, function notation questions ask you to evaluate (find f(3)), compose (find f(g(x))), or interpret (explain what f(5) = 200 means in a given context).
Evaluating a function means substituting the input for the variable everywhere it appears. Students sometimes see f(a + 2) and substitute only into one occurrence of x, or forget to simplify after substituting. For a linear function this is straightforward, but practicing the notation carefully prepares you for the non-linear function questions in the Advanced Math section.
The test frequently gives a linear function in a table or verbal description and asks you to write the function rule. To find the rule from a table, compute the slope from any two rows, then use one row to find b. The question may also ask you to evaluate the function at a value not in the table, which requires the equation rather than reading directly.
Another common question type involves transformations: 'If f(x) = 3x - 2, what is f(x + 4)?' Here you substitute (x + 4) for x everywhere: f(x + 4) = 3(x + 4) - 2 = 3x + 12 - 2 = 3x + 10. The transformed function has the same slope but a different y-intercept, corresponding to a horizontal shift of the graph.
Key Rules & Formulas
Memorize these rules — they come up directly in practice questions.
To evaluate f(a), replace every x in the formula with a and simplify.
If f(x) = 5x - 1, then f(3) = 5(3) - 1 = 14.
A linear function has a constant rate of change (slope) between any two points.
If f(1) = 4 and f(3) = 10, the slope is (10-4)/(3-1) = 3.
The y-intercept f(0) = b is the initial value when the input is zero.
For f(x) = 4x + 7, f(0) = 7 is the starting value.
To find the function rule from a table, compute slope then use y = mx + b with any point.
If (2, 9) and (5, 18) are in the table, m = (18-9)/(5-2) = 3; then 9 = 3(2) + b gives b = 3, so f(x) = 3x + 3.
f(x + k) shifts the graph of f horizontally by k units (left if k > 0).
If f(x) = 2x + 1, then f(x + 3) = 2(x+3) + 1 = 2x + 7.
Linear Functions Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
If f(x) = 7x - 3, what is f(4)?
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Correct answer: A. 25
Explanation
Substitute x = 4: f(4) = 7(4) - 3 = 28 - 3 = 25.
A linear function f satisfies f(0) = 5 and f(2) = 11. What is f(5)?
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Correct answer: B. 20
Explanation
The slope is (11 - 5)/(2 - 0) = 6/2 = 3. With y-intercept 5, f(x) = 3x + 5. So f(5) = 3(5) + 5 = 15 + 5 = 20.
The function g(x) = -2x + 10 models the amount of fuel (in gallons) remaining in a car's tank after x hours of driving. What does g(0) represent in this context?
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Correct answer: C. The car has 10 gallons of fuel at the start
Explanation
g(0) = -2(0) + 10 = 10. Since x = 0 means the start of the trip, g(0) = 10 represents the initial amount of fuel in the tank, which is 10 gallons.
If f(x) = 4x + c and f(3) = 19, what is f(-2)?
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Correct answer: D. -1
Explanation
From f(3) = 19: 4(3) + c = 19 → 12 + c = 19 → c = 7. So f(x) = 4x + 7. Then f(-2) = 4(-2) + 7 = -8 + 7 = -1.
The table below shows values of a linear function h. If h(1) = 8, h(3) = 14, and h(5) = 20, what is h(10)?
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Correct answer: A. 35
Explanation
From the table, the slope is (14-8)/(3-1) = 6/2 = 3. Using h(1) = 8: h(x) = 3x + b → 8 = 3(1) + b → b = 5. So h(x) = 3x + 5. Then h(10) = 3(10) + 5 = 35.
If f(x) = 5x - 3, which of the following is equivalent to f(2x + 1)?
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Correct answer: B. 10x + 2
Explanation
Substitute (2x + 1) for x: f(2x+1) = 5(2x+1) - 3 = 10x + 5 - 3 = 10x + 2.
A phone plan costs $25 per month plus $0.10 per text message. The function C(t) = 0.10t + 25 models the monthly cost C in dollars for t text messages. For what value of t does the monthly cost equal $40?
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Correct answer: C. t = 150
Explanation
Set C(t) = 40: 0.10t + 25 = 40 → 0.10t = 15 → t = 150. The cost equals $40 when 150 text messages are sent.
For the linear function f, f(a) = 12 and f(a + 4) = 28. What is f(a - 3)?
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Correct answer: D. 0
Explanation
The slope is (f(a+4) - f(a))/(4) = (28-12)/4 = 16/4 = 4. So f decreases by 4 for each unit decrease in x. From f(a) = 12, going 3 units left: f(a-3) = 12 - 4(3) = 12 - 12 = 0.
Let f(x) = mx + b. If f(2) - f(0) = 10 and f(0) = 7, what is the value of f(4)?
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Correct answer: A. 27
Explanation
f(2) - f(0) = (2m + b) - b = 2m = 10, so m = 5. Since f(0) = b = 7, the function is f(x) = 5x + 7. Then f(4) = 5(4) + 7 = 27.
A company's weekly profit P (in hundreds of dollars) is modeled by P(n) = 12n - 84, where n is the number of units sold. What is the minimum number of units the company must sell to make a positive profit?
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Correct answer: C. 8
Explanation
We need P(n) > 0: 12n - 84 > 0 → 12n > 84 → n > 7. Since n must be a whole number (units sold), the minimum is n = 8.
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Common Mistakes to Avoid
Use this checklist to catch common traps while working through Linear Functions questions, then add your own patterns as you review missed answers.
- !Substituting the input into only one instance of x when the variable appears multiple times in the formula.
- !Confusing f(x + 2) with f(x) + 2—the first shifts the input, the second shifts the output.
- !Misreading a table: computing the slope as rise/run but using x-values in the numerator and y-values in the denominator.
- !Interpreting f(a) = b as 'f times a equals b' rather than 'the output when input is a is b.'
- !Forgetting to interpret answers in context—'f(10) = 350' might mean '350 dollars after 10 months,' and the question may ask for the unit.
Strategy Tips: Linear Functions
Practice function notation until evaluating f(a) and f(a + k) is completely automatic—these appear in both Algebra and Advanced Math questions.
When building a function from a word problem, identify the independent variable (what you control) and dependent variable (what changes as a result), then find the slope and initial value from the context.
For table-based questions, always check that the rate of change is constant (confirming it is linear) before writing the function rule.
On questions asking about transformations, expand and simplify f(x + k) completely—do not leave it in factored form, as the answer choices will be in simplified form.
Other Algebra Subtopics
Linear Equations in One Variable
Solving equations with a single unknown, from simple one-step problems to multi-step equations with fractions and parentheses.
Linear Equations in Two Variables
Interpreting and writing equations relating two quantities, including converting between equation forms and identifying key features like slope and intercepts.
Systems of Two Linear Equations
Finding the intersection of two lines by substitution or elimination, and analyzing when systems have no solution or infinitely many solutions.
Linear Inequalities in One or Two Variables
Solving and graphing inequalities, interpreting solution sets, and modeling real-world constraints with inequality expressions.
Master Linear Functions
Continue with this skill in the question bank, or test how it holds up under section timing before returning to the surrounding Algebra skills.