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Home›Algebra›Functions

Inverse functions

C = (5/9)(F − 32) turns 75 °F into about 24 °C. Its inverse, F = (9/5)C + 32, turns 26 °C into about 79 °F.

Betty reads a Celsius forecast in Fahrenheit

OpenStax tells of Betty, who is going to Milan and thinks in Fahrenheit. The formula C = (5/9)(F − 32) converts Fahrenheit to Celsius: 75 °F is (5/9)(43), about 24 °C. But the forecast is in Celsius, and solving the equation for F every time would be tedious. What she wants is a second formula that takes a Celsius temperature and returns Fahrenheit. That is an inverse function: the outputs of the first become the inputs of the second, and the inputs become the outputs.

The −1 in f⁻¹ is not an exponent

The inverse of f is written f⁻¹, read "f inverse". The −1 is part of the name, not an exponent: f⁻¹(x) is not 1/f(x). The defining property is that the two functions undo each other. If f(a) = b, then f⁻¹(b) = a, and so f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. For example, if f(3) = 10, then f⁻¹(10) = 3 without any formula at all. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

Solve for x, then swap the letters

To find an inverse from a formula, write y = f(x), solve for x, and then swap the names of the letters. For f(x) = 2x + 1: y = 2x + 1, so y − 1 = 2x and x = (y − 1)/2. Renaming gives f⁻¹(x) = (x − 1)/2. Check it: f(3) = 7 and f⁻¹(7) = (7 − 1)/2 = 3. For Betty, solving C = (5/9)(F − 32) for F gives F = (9/5)C + 32, so 26 °C becomes 26 × 9/5 + 32 = 78.8, about 79 °F, and 20 °C becomes 68 °F. Undo the steps in reverse order: f doubles and then adds 1, so f⁻¹ subtracts 1 and then halves.

x² sends 3 and −3 to the same 9

Not every function can be reversed. f(x) = x² sends both 3 and −3 to 9, so a reverse machine fed 9 would not know which to return. A function has an inverse exactly when it is one-to-one, which you can check with the horizontal line test from the first unit. OpenStax's fix for x² is to restrict its domain to [0, ∞). On that half the function is one-to-one, and its inverse is the square root function, √x. That is why √9 means 3 and not −3.

The graph of f⁻¹ is f reflected in the line y = x

Swapping inputs and outputs swaps the coordinates of every point: (a, b) on f becomes (b, a) on f⁻¹. Geometrically, that swap is a reflection across the diagonal line y = x. So the graph of an inverse is the mirror image of the original in that line. For f(x) = 2x + 1, the point (1, 3) mirrors to (3, 1), which is on y = (x − 1)/2. In the lab, move the point along f and watch its partner move along f⁻¹. Choose f(x) = x² on all real numbers and the mirror image turns into a sideways parabola that fails the vertical line test.

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f(x) = 2x + 1 (blue), y = x (dashed) and f⁻¹(x) = (x − 1)/2 (orange). The point (1, 3) and its mirror (3, 1). Try x³, then x² on all real numbers.
  1. What is the inverse of f(x) = 2x + 1?
    • (x − 1)/2
    • 2x − 1
    • 1/(2x + 1)
    • (x + 1)/2
  2. For f(x) = 2x + 1, what is f⁻¹(7)?
  3. If f(3) = 10, what is f⁻¹(10)?
  4. For f(x) = x³, what is f⁻¹(8)?
  5. Does f(x) = x², on all real numbers, have an inverse function?
    • No, it is not one-to-one
    • Yes, √x
    • Yes, 1/x²
    • Yes, −x²
  6. What does f⁻¹(x) mean?
    • The inverse function of f at x
    • 1/f(x)
    • f(x) to the power −1, as a number
    • −f(x)
  7. (2, 5) is on the graph of f. Which point is on the graph of f⁻¹?
    • (5, 2)
    • (2, 5)
    • (−2, −5)
    • (1/2, 1/5)
  8. Convert 20 °C to Fahrenheit with F = (9/5)C + 32.
  9. For f(x) = x/2 − 3, what is f⁻¹(1)?
  10. Which is always true for a function f and its inverse?
    • f(f⁻¹(x)) = x
    • f(x) · f⁻¹(x) = 1
    • f⁻¹(x) = −f(x)
    • f(x) = f⁻¹(x)
Inverse function f⁻¹Swaps inputs and outputs: f(a) = b means f⁻¹(b) = a
Finding f⁻¹Write y = f(x), solve for x, rename the letters
Which functions have inversesOne-to-one functions (horizontal line test)
Graph of f⁻¹The mirror image of f in the line y = x
Common mistakes
  • Writing f⁻¹(x) = 1/(3x − 6) for f(x) = 3x − 6. That is the reciprocal 1/f(x). Undo the steps instead: f⁻¹(x) = (x + 6)/3. Check with 4: f(4) = 6 and f⁻¹(6) = 4, while 1/f(4) = 1/6.
  • Undoing f(x) = 2x + 1 as "halve, then subtract 1", which gives x/2 − 1. f doubles and then adds 1, so the inverse goes backwards: subtract 1 first, then halve. At 7, x/2 − 1 gives 2.5, but f(2.5) = 6, not 7.
  • Reflecting f over the x-axis to draw f⁻¹. The x-axis mirror gives −f(x). The inverse swaps coordinates, which is the mirror in y = x: (2, 7) on f(x) = 3x + 1 goes to (7, 2), not (2, −7).
Teacher's noteTwo spots cause trouble. The notation f⁻¹ looks like a power, so spend a minute on one concrete function and compute f⁻¹(5) next to 1/f(5). And "swap x and y" sounds like a trick. Build a table of values for f first, swap the two columns, and only then solve for x, so the swap has a reason. Leave the restricted x² case until the one-to-one idea is settled.
Key points
  • f⁻¹ undoes f: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x.
  • f⁻¹(x) is not 1/f(x).
  • To find it: y = f(x), solve for x, swap the letters.
  • Only one-to-one functions have inverses; the graph of f⁻¹ is f reflected in y = x.
Sources for this course (8)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 중학교 1~3학년군 [9수02-06] 그래프 해석(증가와 감소), [9수02-14] 함수의 개념·함숫값, 용어 함수·함숫값·기울기·평행이동, 고려 사항 「한 양이 변함에 따라 다른 양이 하나씩 정해지는 두 양 사이의 대응 관계를 이용하여 도입」; 고등학교 공통수학2 [10공수2-01-06] 평행이동, [10공수2-01-07] 원점·x축·y축·직선 y = x에 대한 대칭이동, [10공수2-03-01] 함수의 개념(두 집합 사이의 대응 관계), [10공수2-03-02] 합성함수, [10공수2-03-03] 역함수, 용어 정의역·치역·공역·대응·일대일대응·일대일함수·합성함수·역함수, 기호 f: X → Y, g ∘ f, (g ∘ f)(x), y = g(f(x)), f⁻¹, y = f⁻¹(x), 교수·학습 용어 조건제시법; 미적분Ⅰ 용어 구간·닫힌구간·열린구간 [a, b], (a, b), 평균변화율, [12미적Ⅰ-02-07] 증가와 감소, 극댓값·극솟값)
  • Textbook: OpenStax, College Algebra 2e, 3.1 Functions and Function Notation (relation, domain and range of ordered pairs, {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}, the odd/even relation that is not a function, coffee menu prices, vertical line test, horizontal line test, one-to-one: openstax.org/books/college-algebra-2e/pages/3-1-functions-and-function-notation)
  • Textbook: OpenStax, College Algebra 2e, 3.2 Domain and Range (no even root of a negative number, no division by 0, interval notation with brackets and parentheses, (0, 100] for "more than 0 and at most $100", {(2, 10), (3, 10), (4, 20), (5, 30), (6, 40)}: .../pages/3-2-domain-and-range)
  • Textbook: OpenStax, College Algebra 2e, 3.3 Rates of Change and Behavior of Graphs (average rate of change, increasing and decreasing intervals, local maxima and minima: .../pages/3-3-rates-of-change-and-behavior-of-graphs)
  • Textbook: OpenStax, College Algebra 2e, 3.4 Composition of Functions (f(g(x)); f(x) = 2x + 1 and g(x) = 3 − x give f(g(x)) = 7 − 2x and g(f(x)) = −2x + 2, so composition is not commutative; c(s) calories and s(t) sit-ups; f(x) = 1/x and g(x) = x − 3: .../pages/3-4-composition-of-functions)
  • Textbook: OpenStax, College Algebra 2e, 3.5 Transformation of Functions (g(x) = f(x) + k vertical shift, f(x − h) horizontal shift, −f(x) and f(−x) reflections, af(x) stretches and compressions, roof vent example S(t) = V(t) + 20 (open vent area in square feet plus 20): .../pages/3-5-transformation-of-functions)
  • Textbook: OpenStax, College Algebra 2e, 3.7 Inverse Functions (f⁻¹ is not an exponent and not 1/f, C = (5/9)(F − 32), 75 °F is about 24 °C, 26 °C is about 79 °F, x² restricted to [0, ∞), reflection about y = x: .../pages/3-7-inverse-functions)
  • History block only: MacTutor, University of St Andrews, The function concept (Leibniz, August 1673, the word "function"; Johann Bernoulli's letter of 2 September 1694; Euler, Introductio in analysin infinitorum 1748 and Institutiones calculi differentialis 1755; Dirichlet 1837; Goursat 1923 "y = f(x)": mathshistory.st-andrews.ac.uk/HistTopics/Functions)

Checked: 2026-10-18