5
17
1
1
0
5
17
1
1
0
The Inverse Function Calculator finds the input value $$x$$ that produces a given output $$y$$ for common function types — effectively computing $$f^{-1}(y)$$, the inverse function. If a function $$f$$ maps $$x$$ to $$y$$, then its inverse $$f^{-1}$$ maps $$y$$ back to $$x$$. Inverse functions are a cornerstone concept in mathematics, appearing throughout algebra, calculus, and applied sciences whenever you need to "undo" a mathematical operation or solve for an unknown input.
The concept is deceptively simple: given $$y = f(x)$$, solve for $$x$$ in terms of $$y$$. For a linear function $$f(x) = ax + b$$, the inverse is $$f^{-1}(y) = \frac{y - b}{a}$$. For an exponential function $$f(x) = ae^x + b$$, the inverse involves the natural logarithm: $$f^{-1}(y) = \ln\left(\frac{y - b}{a}\right)$$. Each function family has its own inverse pattern, and this calculator handles the five most important types: linear, quadratic (restricted to non-negative $$x$$), square root, exponential, and logarithmic.
Not every function has an inverse. A function must be one-to-one (injective) — each output corresponds to exactly one input — to have an inverse. The standard quadratic $$f(x) = ax^2 + b$$ is not one-to-one over all reals because both $$x$$ and $$-x$$ give the same output. By restricting the domain to $$x \geq 0$$, we obtain a one-to-one function whose inverse is $$f^{-1}(y) = \sqrt{\frac{y - b}{a}}$$. This domain restriction is handled automatically in the calculator.
Inverse functions have a beautiful geometric property: the graph of $$f^{-1}$$ is the reflection of the graph of $$f$$ across the line $$y = x$$. This means that the point $$(a, b)$$ on the graph of $$f$$ corresponds to the point $$(b, a)$$ on the graph of $$f^{-1}$$. The composition property $$f(f^{-1}(y)) = y$$ and $$f^{-1}(f(x)) = x$$ is the defining characteristic — and the calculator includes a verification output that computes $$f(f^{-1}(y))$$ to confirm it equals your input $$y$$.
In applications, inverse functions appear everywhere. In finance, if compound interest is modeled by an exponential, the inverse (logarithm) tells you how long until an investment reaches a target value. In physics, if position is a function of time, the inverse tells you when an object reaches a specific position. In statistics, the inverse of the cumulative distribution function gives you critical values for hypothesis testing. In engineering, inverse functions are used to calibrate sensors, decode signals, and invert system models.
This calculator provides the inverse value, a verification check, the y-intercept of the original function, and the value of the inverse at 0 — giving you a comprehensive view of both the function and its inverse at key reference points.
Select the function type, enter coefficients $$a$$ and $$b$$, and provide the $$y$$-value for which you want to find $$x = f^{-1}(y)$$. The calculator solves for $$x$$ using the appropriate inverse formula:
The verification output computes $$f(f^{-1}(y))$$ to confirm it equals $$y$$. A result of 0 in cases with domain restrictions indicates the input is outside the valid range.
The primary output $$f^{-1}(y)$$ is the $$x$$-value that produces the given $$y$$-value when plugged into the function. The verification field should match your input $$y$$ — any discrepancy indicates a domain issue or numerical rounding. The value $$f(0)$$ is the y-intercept of the original function, and $$f^{-1}(0)$$ is the x-intercept (root) of the original function. If a result shows 0 unexpectedly, check whether the input falls outside the function's range.
Inputs
Results
f⁻¹(17) = (17 − 2)/3 = 15/3 = 5. Verify: f(5) = 3(5) + 2 = 17. The y-intercept is f(0) = 2, and the x-intercept (root) is f⁻¹(0) = −2/3.
Inputs
Results
f⁻¹(55.6) = ln((55.6 − 1)/2) = ln(27.3) ≈ 3.3083. Verify: f(3.3083) = 2e^3.3083 + 1 ≈ 55.6. The inverse at 0 is ln(−1/2) which is undefined; the calculator shows f⁻¹(0) for y=0 as ln(−0.5) which is invalid, showing 0.
An inverse function $$f^{-1}$$ reverses the operation of a function $$f$$. If $$f(a) = b$$, then $$f^{-1}(b) = a$$. It "undoes" what the original function does. For example, if $$f(x) = 2x + 3$$, then $$f^{-1}(y) = \frac{y-3}{2}$$ reverses the doubling and adding.
No. Only one-to-one (injective) functions have inverses. A function is one-to-one if each output comes from exactly one input. The horizontal line test checks this: if any horizontal line intersects the graph more than once, the function is not one-to-one and has no inverse (unless the domain is restricted).
The function $$f(x) = ax^2 + b$$ is not one-to-one over all reals because $$f(x) = f(-x)$$. Restricting to $$x \geq 0$$ makes it one-to-one, allowing the inverse $$f^{-1}(y) = \sqrt{(y-b)/a}$$ to return a unique non-negative value.
The verification computes $$f(f^{-1}(y))$$, which should equal your original $$y$$. This confirms the inverse was computed correctly. Any small discrepancy is due to floating-point rounding. A result of 0 when your $$y$$ was non-zero suggests a domain violation.
Because $$\ln(e^x) = x$$ and $$e^{\ln x} = x$$. The exponential function and the natural logarithm undo each other. This is why the inverse of $$f(x) = ae^x + b$$ involves $$\ln$$, and the inverse of $$f(x) = a\ln x + b$$ involves $$e$$.
Reflect the graph of $$f$$ across the line $$y = x$$. Equivalently, swap the $$x$$ and $$y$$ coordinates of every point on the original graph. If $$(2, 5)$$ is on the graph of $$f$$, then $$(5, 2)$$ is on the graph of $$f^{-1}$$.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
