Inverse Function Calculator - Step-by-Step Solutions

Enter a function of $x$. The calculator finds its inverse function $f^{-1}(x)$, shows step by step how the equation ${y = f(x)}$ is solved, and finds the domain and the range of both functions. If the function is not one-to-one, it finds the inverse on each part of the domain where it is one-to-one.

How to enter a function

• Write powers with ^: x^2 is $x^2$, e^(2x) is $e^{2x}$ and 2^x is $2^x$.
• The multiplication sign can be left out: 2x, 3(x+1), 2sin(3x). You can also write 2*x.
• (2x+1)/(x-3) is $\frac{2x+1}{x-3}$: put the numerator and the denominator in brackets.
• sqrt(x) is $\sqrt{x}$, cbrt(x) or x^(1/3) is $\sqrt[3]{x}$, and root(x, 4) is $\sqrt[4]{x}$.
• ln(x) is the natural logarithm, log(x) is the logarithm to base 10 and log_2(x) is $\log_2 x$.
• sin(x), cos(x), tan(x), cot(x), arcsin(x), arccos(x), arctan(x), sinh(x), cosh(x), tanh(x).
• |x| is the absolute value, pi is $\pi$ and e is the number $e$.
• You can also type f(x) = ... or y = .... The function may contain one letter only, its variable.

Examples: $2x + 3$ $\frac{2x + 1}{x - 3}$ $x^2 - 4x + 1$ $\sqrt{x - 1}$ $3e^{2x - 1} + 4$ $\log_2(x + 3)$ $\frac{e^x - 1}{e^x + 1}$

How to find the inverse of a function

The inverse function $f^{-1}$ undoes what $f$ does: if ${f(a) = b}$, then ${f^{-1}(b) = a}$. To find it:

1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve the equation for $y$.
4. Replace $y$ with $f^{-1}(x)$.

For ${f(x) = 2x + 3}$ these steps give ${x = 2y + 3}$, then ${2y = x - 3}$, so ${f^{-1}(x) = \frac{x - 3}{2}}$. You can also solve ${y = f(x)}$ for $x$ first and swap the letters at the end: the inverse is the same.

When does a function have an inverse?

Only a one-to-one function has an inverse: each of its values must come from exactly one $x$. A function that increases on its whole domain, or decreases on it, is one-to-one. On a graph, every horizontal line crosses a one-to-one function at most once.

${f(x) = x^2}$ is not one-to-one, because ${f(-2) = f(2) = 4}$. Restricted to ${x \geq 0}$ it is one-to-one, and its inverse is $\sqrt{x}$; restricted to ${x \leq 0}$ the inverse is $-\sqrt{x}$. The calculator finds such parts of the domain by itself. For the trigonometric functions it gives the inverse on the interval of the principal values, as $\arcsin$, $\arccos$ and $\arctan$ do.

The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$. The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line ${y = x}$.

Inverses of basic functions

Methods with examples

The calculator solves ${y = f(x)}$ the way a textbook does. Click an example to see the method at work.

1. Linear functions. Undo the operations in reverse order: subtract, then divide.
   ${y = ax + b} \;\Rightarrow\; {x = \frac{y - b}{a}}$
   Example: $5x - 7$
2. Linear fractional functions. Multiply both sides by the denominator, move the terms with $x$ to one side and take $x$ out of the brackets.
   ${y = \frac{ax + b}{cx + d}} \;\Rightarrow\; {x = \frac{b - dy}{cy - a}}$
   Examples: $\frac{2x + 1}{x - 3}$, $\frac{x}{x + 1}$
3. Quadratic functions. Complete the square and take the square root of both sides. A quadratic function is not one-to-one: its inverse has the sign + on one side of the vertex and the sign − on the other. A function such as ${x + \frac{1}{x}}$ also leads to a quadratic equation.
   ${ax^2 + bx + c} = {a(x - h)^2 + k}$
   Examples: $x^2 - 4x + 1$, $2x^2 + 8x - 5$, $x + \frac{1}{x}$
4. Powers and roots. A power is undone by a root and a root by a power. A square root is never negative, so the inverse of $\sqrt{x - 1}$ is ${x^2 + 1}$ only for ${x \geq 0}$.
   ${\sqrt[3]{x} = y} \;\Rightarrow\; {x = y^3}$
   Examples: $\sqrt{2x + 3} - 4$, $(x - 1)^3 + 4$
5. Exponential and logarithmic functions. Take the logarithm of both sides, or write a logarithm as a power.
   ${a^x = y} \;\Leftrightarrow\; {x = \log_a y}$
   Examples: $3e^{2x - 1} + 4$, $\log_2(x + 3)$, $2^{x - 1} + 3$
6. Trigonometric functions. Apply the inverse trigonometric function. It gives the principal value, so the inverse is the one of $f$ restricted to an interval.
   ${\sin x = y},\; {-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}} \;\Rightarrow\; {x = \arcsin y}$
   Examples: $2\sin 3x + 1$, $\arctan 2x$
7. Substitution. When $x$ occurs several times, but always in the same expression such as $e^x$ or $\sqrt{x}$, call that expression $u$, solve for $u$ and then for $x$. Examples: $\frac{e^x - 1}{e^x + 1}$, $e^{2x} - 2e^x$

Other resources on functions

Inverse function - definition and examples Domain and range of functions Operations on functions Logarithms