Visualize the inverse of any function by reflecting over the line y=x. Enter your function, choose a domain, and instantly see the original and its inverse. Enhanced with algebraic inverse display and numerical sample table.
An inverse function reverses the operation of the original function. If y = f(x), then the inverse, denoted f−1(y), satisfies f−1(f(x)) = x and f(f−1(y)) = y. Graphically, the inverse is the mirror image of the original function across the line y = x.
A function must be one‑to‑one (injective) to have an inverse over its entire domain. If not, we restrict the domain to make it one‑to‑one.
The concept of inverse functions dates back to the development of logarithms by John Napier in the 17th century (inverse of exponentials). Later, the formal definition was refined by mathematicians like Leibniz and Euler. Inverse functions are fundamental in solving equations, cryptography, signal processing, and economics (e.g., demand and supply functions). The ability to visualize the inverse helps in understanding transformations and the idea of reversibility in mathematics.
y = f(x).
x = f(y).
y in terms of x.
f−1(x).
Example: For f(x) = 2x + 3, we set y = 2x + 3, swap → x = 2y + 3, solve → y = (x - 3)/2. So f−1(x) = (x-3)/2.
f−1 is the range of f.
f−1 is the domain of f.
f is not one‑to‑one on the chosen interval, the reflection will not pass the vertical line test – in that case we restrict the domain further.
Our tool automatically detects whether the function appears one‑to‑one on the given domain by checking for duplicate y‑values. If not, we issue a warning and still show the reflected relation.
| Function | Interval | Monotonicity | Sample f⁻¹(y) | Graphical symmetry |
|---|---|---|---|---|
| f(x) = 2x+1 | [-3,3] | Increasing | f⁻¹(3) = 1.0 | Line, perfect mirror |
| f(x) = x² | [0,4] | Increasing | f⁻¹(4) = 2.0 | Parabola & sqrt |
| f(x) = eˣ | [-2,2] | Increasing | f⁻¹(2) ≈ 0.6931 (ln 2) | Exponential & log |
| f(x) = x³ | [-2,2] | Increasing | f⁻¹(8) = 2.0 | Cubic & cube root |
A thermistor has resistance R that varies with temperature T according to R = 1000 * exp(3500/(T+273)) - 500. To measure temperature from resistance, engineers need the inverse function. Using numerical methods, they create a lookup table from the measured resistance. Our graphical tool helps visualize this relationship and verify monotonicity.
x² is not √x over all reals: It is only valid for x ≥ 0. The inverse of the full parabola is a relation, not a function.
f−1(x) does not mean 1/f(x): The notation is often confused with reciprocal; it means inverse function, not multiplicative inverse.
We use math.js to parse and evaluate your function. The graph is drawn using Chart.js. For the inverse, we sample points (x, f(x)) and then plot (f(x), x) after sorting by the new x‑coordinate. This effectively shows the reflected curve. The dashed grey line is y=x. The tool also approximates the range of the function and checks for duplicate y‑values to assess one‑to‑oneness. Additionally, we attempt to derive an algebraic inverse for simple patterns (linear, power, exp/log) and display it both in the properties table and in a separate box.
^ (e.g., x^2) or ** (e.g., x**3). For square root, use sqrt(x). Exponential: exp(x). Logarithm: log(x) (natural log) or log10(x).
x² from -5 to 5), the reflected graph will fail the vertical line test. That means the inverse is a relation, not a function. You should restrict the domain to make it one‑to‑one (e.g., x ≥ 0).
sin(x), cos(x), tan(x). They are evaluated in radians. For inverses, use asin(x), acos(x), atan(x). Remember that sine and cosine are not one‑to‑one over the whole real line; restrict the domain (e.g., x from -1.5 to 1.5 for a monotonic piece).