What is an inverse function?

An inverse function reverses what the original function does. If a function f takes an input x and outputs y, we write y = f(x). The inverse, written as f⁻¹, takes that output value and sends it back to the input: x = f⁻¹(y). In other words, the inverse “undoes” the function.

This idea shows up everywhere in algebra and precalculus because it connects directly to solving equations. If you can describe a process as a function (multiply, shift, square, exponentiate), then the inverse is the “reverse” process (divide, shift back, square-root, logarithm). Inverses are also why you learn paired operations like exponentials and logs, or squaring and square roots.

One of the most important rules is the composition identity: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, as long as you stay in the correct domains. That’s not just a definition — it’s a powerful self-check. If your inverse is right, composing the function and its inverse should bring you back to where you started.

• Inverse function is written f⁻¹(x) and means the “undo” function.
• Reciprocal is 1/f(x). That’s a different operation.
• Negative exponent (like x⁻¹) means 1/x, not an inverse function unless f(x)=x.

How to find an inverse (the classic 4-step method)

Most textbook inverse problems use the same strategy. You start with y = f(x), then you swap the roles of input and output, and finally solve for the new output variable. Here’s the method you can memorize:

1. Write y = f(x).
2. Swap x and y (because the inverse flips input and output).
3. Solve for y.
4. Rename y as f⁻¹(x).

Example (linear): If f(x)=2x+3, start with y=2x+3. Swap: x=2y+3. Solve for y: y=(x−3)/2. So f⁻¹(x)=(x−3)/2. If you plug it back in, f(f⁻¹(x)) = 2·((x−3)/2)+3 = x. Perfect.

The only “gotcha” is that an inverse exists only when the function is one-to-one. That’s why you often see “restrict the domain” for quadratics (and even-power functions). The calculator above lets you choose a branch so the inverse is truly a function.

Inverse formulas for the most common function families

When you recognize a function family, you can often write its inverse almost instantly. Below are the exact forms used in this calculator, along with the inverse you’ll get. These are extremely common in algebra, precalc, and early calculus.

• Linear: f(x)=a·x+b → f⁻¹(x)=(x−b)/a (requires a≠0).
• Quadratic (vertex form): f(x)=a(x−h)²+k → f⁻¹(x)=h ± √((x−k)/a). You must pick a branch (x≥h or x≤h).
• Reciprocal: f(x)=a/(x−h)+k → f⁻¹(x)=h + a/(x−k) (requires x≠k).
• Power: f(x)=a(x−h)ⁿ+k → f⁻¹(x)=h + ((x−k)/a)^(1/n). If n is even, you usually need a domain restriction like x≥h.
• Exponential: f(x)=a·b^(x−h)+k → f⁻¹(x)=h + log_b((x−k)/a). Requires a≠0, b>0, b≠1, and (x−k)/a > 0.
• Logarithmic: f(x)=a·log_b(x−h)+k → f⁻¹(x)=h + b^((x−k)/a). Requires a≠0, b>0, b≠1.

Notice the “swap pairs”: exponentials invert to logs, and logs invert to exponentials. Squares invert to square roots (with a sign choice based on the branch), and multiplying by a becomes dividing by a.

Worked examples you can copy for homework

Example 1: Linear

Let f(x)=−3x+12. Write y=−3x+12. Swap: x=−3y+12. Solve: −3y=x−12 → y=(12−x)/3 = 4 − x/3. So f⁻¹(x)=4 − x/3.

Example 2: Quadratic with restriction

Let f(x)=(x−1)²+5 with domain x≥1. Write y=(x−1)²+5. Swap: x=(y−1)²+5. Solve: x−5=(y−1)² → √(x−5)=y−1 → y=1+√(x−5). Because we restricted x≥1, we choose the “+” branch, making the inverse a true function.

Example 3: Exponential

Let f(x)=2·3^(x−4)−7. Write y=2·3^(x−4)−7. Swap: x=2·3^(y−4)−7. Solve: x+7=2·3^(y−4) → (x+7)/2=3^(y−4). Take log base 3: log₃((x+7)/2)=y−4 → y=4+log₃((x+7)/2). So f⁻¹(x)=4+log₃((x+7)/2).

Inverse Function Calculator FAQs

• How do I know if a function has an inverse?

  A function has an inverse function if it’s one-to-one, meaning different inputs always give different outputs. Graphically, that’s the horizontal line test. If a horizontal line hits the graph more than once, the function is not one-to-one unless you restrict the domain.

• Why do quadratics need a branch?

  A parabola repeats y-values (for most y you get two x’s). That breaks one-to-one behavior. Restricting to x≥h (right side) or x≤h (left side) makes it one-to-one, so the inverse becomes a function.

• What does “domain and range swap” mean?

  If the domain of f is the set of allowed x-values, then the inverse’s domain becomes the original range. Similarly, the inverse’s range becomes the original domain. Swapping x and y is basically swapping those sets.

• Is f⁻¹(x) the same as 1/f(x)?

  No. f⁻¹(x) is the inverse function, while 1/f(x) is the reciprocal. They only match in special cases.

• How can I verify an inverse quickly?

  Plug the inverse into the original: compute f(f⁻¹(x)). If it simplifies to x (on the correct domain), you’ve got the correct inverse. You can also test with a number: pick an x, compute y=f(x), then compute f⁻¹(y). You should get your original x back.

• Why are there restrictions for logs and exponentials?

  Logs need a positive argument: x−h > 0. Exponentials require base b>0 and b≠1 (otherwise it’s not invertible). The calculator warns you when inputs break these rules.

How to get faster at inverse problems

• Spot the family: If it’s a “shift + scale” of a known function, the inverse is too.
• Undo in reverse order: If f does “+k then ×a then exponent,” inverse does “log then ÷a then −k”.
• Watch one-to-one: If the graph isn’t one-to-one, add a domain restriction before inverting.
• Keep domain notes: Most mistakes are sign/branch errors or missing domain restrictions.
• Practice a quick check: One numeric test can save you from a wrong sign.

If you want a viral classroom trick: screenshot the inverse and the verification line (f(f⁻¹(x))=x), then share it in group chats — it’s the fastest “yes, that’s right” proof.

Common inverse mistakes (and how to avoid them)

Most “wrong inverses” happen for the same predictable reasons. First, people forget that f⁻¹ is a function name, not a power. Writing f⁻¹(x) does not mean 1/f(x). Second, students often swap x and y but forget to finish the job by solving for y. A good habit is to circle the final line and make sure it reads “y = …” before you rename it f⁻¹(x).

Another big error is ignoring domain restrictions. Quadratics and even-power functions can’t be one-to-one on the entire real line, so the inverse requires a branch choice (right side or left side). If your teacher says “find the inverse,” they usually mean “find the inverse function,” which implicitly requires a restricted domain. The same idea applies to logs: the inside must be positive, so x−h>0. And with exponentials/logs, the base must be valid (b>0 and b≠1), otherwise there is no meaningful inverse.

Finally, always do a quick sanity check. Pick a simple input x (like 0 or 1), compute y=f(x), then compute x₂=f⁻¹(y). If x₂ matches your original x, you’re almost certainly correct. This 10-second test is also extremely shareable: it’s a “receipt” that your inverse works.

How inverses look on a graph (easy mental picture)

If you graph a function and its inverse on the same coordinate plane, they’re mirror images across the line y = x. That mirror idea is why the inverse method starts by swapping x and y: you’re literally swapping the roles of horizontal and vertical axes.

This helps when you’re unsure about the “+” or “−” sign in a square-root inverse. If your original function uses only the right branch (x≥h), then the inverse must output values ≥h as well. On a graph, that means the inverse lives on the correct mirrored side. When in doubt, graph a couple points and reflect them across y=x.