ScoreReady SAT
Drill by score band: 400–500 500–600 600–700 700–800

Function Notation and Composition · Sub-skill drill

Inverse Functions

The inverse f^(−1)(x) of a function f(x) undoes f. Algebraically, swap x and y in the equation y = f(x), then solve for y; the resulting expression is the inverse. Graphically, the inverse is the reflection of the original function across the line y = x. SAT questions test whether you can find an inverse algebraically or recognize the graphical relationship. The check is f(f^(−1)(x)) = x for all x in the domain.

How this sub-skill is tested on the SAT

The inverse f^(−1)(x) of a function f(x) undoes f. Algebraically, swap x and y in the equation y = f(x), then solve for y; the resulting expression is the inverse. Graphically, the inverse is the reflection of the original function across the line y = x. SAT questions test whether you can find an inverse algebraically or recognize the graphical relationship. The check is f(f^(−1)(x)) = x for all x in the domain.

This sub-skill sits inside the broader Function Notation and Composition topic, which is part of the College Board's Passport to Advanced Math content domain. Passport to Advanced Math is the SAT's bridge to the kind of algebraic manipulation you will see in a college precalculus or calculus course. The questions are not about memorizing identities — they are about fluency with structure. Can you factor a quadratic by inspection? Can you read the vertex of a parabola off its standard form? Can you simplify a rational expression without losing a domain restriction? Can you translate between f(x), a graph, and a table of values without panicking? Most students who plateau between 650 and 720 plateau here, because the section rewards algebraic intuit

Practice questions in this drill set

Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.

  1. 400-500 easy

    If f(x) = 4x - 7) and g(x) = 2x + (4), what is the value of f(g(2))?

    1. A 26
    2. B 25
    3. C 1
    4. D 8
    Worked solution

    Answer: B — 25

    Evaluate inside-out. First g(2) = 2(2) + (4) = 8. Then f(8) = 4(8) + (-7) = 25.

    Open question q0353 in its own page →

  2. 500-600 easy

    If f(x) = 2x + (2) and g(x) = 1x + (3), what is the value of f(g(-1))?

    1. A 6
    2. B 7
    3. C 2
    4. D 0
    Worked solution

    Answer: A — 6

    Evaluate inside-out. First g(-1) = 1(-1) + (3) = 2. Then f(2) = 2(2) + (2) = 6.

    Open question q0357 in its own page →

  3. 500-600 medium

    If f(x) = 4x + (7) and g(x) = 4x - 4), what is the value of f(g(4))?

    1. A 23
    2. B 56
    3. C 55
    4. D 12
    Worked solution

    Answer: C — 55

    Evaluate inside-out. First g(4) = 4(4) + (-4) = 12. Then f(12) = 4(12) + (7) = 55.

    Open question q0361 in its own page →

  4. 600-700 medium

    If f(x) = 2x + (3) and g(x) = 3x + (3), what is the value of f(g(-1))?

    1. A 4
    2. B 1
    3. C 0
    4. D 3
    Worked solution

    Answer: D — 3

    Evaluate inside-out. First g(-1) = 3(-1) + (3) = 0. Then f(0) = 2(0) + (3) = 3.

    Open question q0365 in its own page →

  5. 600-700 hard

    If f(x) = 2x + (5) and g(x) = 2x + (1), what is the value of f(g(3))?

    1. A 19
    2. B 11
    3. C 7
    4. D 20
    Worked solution

    Answer: A — 19

    Evaluate inside-out. First g(3) = 2(3) + (1) = 7. Then f(7) = 2(7) + (5) = 19.

    Open question q0369 in its own page →

  6. 700-800 hard

    If f(x) = 5x + (6) and g(x) = 4x - 1), what is the value of f(g(4))?

    1. A 81
    2. B 26
    3. C 82
    4. D 15
    Worked solution

    Answer: A — 81

    Evaluate inside-out. First g(4) = 4(4) + (-1) = 15. Then f(15) = 5(15) + (6) = 81.

    Open question q0373 in its own page →

Why this band assignment matters

Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.

Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.

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