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Home›Algebra

Functions

Six units on functions: what makes a rule a function, domain and range, reading a graph, shifting and stretching graphs, putting one function inside another, and running a function backwards. Each unit has a function lab, from the vertical line test to a mirror in the line y = x.

Progress

Before you start

Most examples are read from graphs and solved with simple equations, so Graphs of lines and Linear equations come first. The transformations unit shifts the parabola y = x², and the inverse unit reuses the Celsius and Fahrenheit formula, so having seen both before helps.

When you finish

Afterwards you can use the vertical line test, find f(4) from a rule, and state that √(x + 2) has a domain starting at −2. You will read rates from a rain barrel graph, move y = x² to vertex (3, 1), show that f(g(x)) and g(f(x)) differ, and find an inverse by solving for x and swapping, as in turning 26 °C into about 79 °F.

  1. 1What is a functionEach number from 1 to 5 paired with its double gives five pairs, from (1, 2) to (5, 10). Because every input gets exactly one output, the pairing is a function.
  2. 2Domain and rangef(x) = √(x + 2) has no real output at x = −3, because −3 + 2 = −1 and a negative number has no real square root. Its domain starts at −2.
  3. 3Reading a graphA rain barrel graph rises from 20 to 60 liters in two hours, so the water came in at an average of 20 liters an hour.
  4. 4Transformations of graphsThe graph of y = (x − 3)² + 1 is the graph of y = x² moved 3 to the right and 1 up. The vertex goes from (0, 0) to (3, 1).
  5. 5Composition of functionsWith f(x) = 2x + 1 and g(x) = 3 − x, f(g(x)) = 7 − 2x but g(f(x)) = 2 − 2x. The order changes the answer.
  6. 6Inverse functionsC = (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.
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