Inverse Functions
Aligned to F-BF.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
An inverse function undoes another: swap x and y, then solve for y. Two functions f and g are inverses exactly when f(g(x)) = x and g(f(x)) = x.
Worked examples
Example 1
Let f(x) = 2x + 6. Swap to get x = 2y + 6, so y = (x − 6)/2. Check: f((x − 6)/2) = 2 × (x − 6)/2 + 6 = (x − 6) + 6 = x, so it is verified.
Example 2
Celsius to Fahrenheit is F(c) = (9/5)c + 32. Find the inverse. Swap: c = (9/5)f + 32, so c − 32 = (9/5)f and f = (5/9)(c − 32). That is the familiar Fahrenheit-to-Celsius formula — the two conversions are inverse functions of each other, which is why applying both returns the original temperature.
Example 3
Common mistake: assuming every function has an inverse function. f(x) = x² fails the horizontal-line test: the line y = 4 hits it at both x = 2 and x = −2, so an inverse would have to send 4 to two places. The domain must be restricted first — on x ≥ 0 the inverse is √x. Any function that repeats an output needs a restriction before it can be inverted.
Practice questions
- Find the inverse of f(x) = x − 7.
- Find the inverse of f(x) = 3x.
- Verify that f(x) = x^3 and g(x) = √[3]{x} are inverses by composition.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Funciones inversas
Una función inversa deshace a otra: intercambia x y y, luego despeja y. Dos funciones f y g son inversas exactamente cuando f(g(x)) = x y g(f(x)) = x.
Ejemplo 1: Sea f(x) = 2x + 6. Intercambia para obtener x = 2y + 6, así que y = (x − 6)/2. Comprueba: f((x − 6)/2) = 2 × (x − 6)/2 + 6 = (x − 6) + 6 = x, entonces queda verificado.
Ejemplo 2: De Celsius a Fahrenheit la fórmula es F(c) = (9/5)c + 32. Halla la inversa. Intercambia: c = (9/5)f + 32, así que c − 32 = (9/5)f y f = (5/9)(c − 32). Esa es la conocida fórmula de Fahrenheit a Celsius — las dos conversiones son funciones inversas entre sí, y por eso aplicar ambas devuelve la temperatura original.
Ejemplo 3: Error común: suponer que toda función tiene función inversa. f(x) = x² falla la prueba de la recta horizontal: la recta y = 4 la corta en x = 2 y en x = −2, así que una inversa tendría que mandar el 4 a dos lugares. Primero hay que restringir el dominio — en x ≥ 0 la inversa es √x. Toda función que repite una salida necesita una restricción antes de poder invertirse.
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