Formula used
The main rule is simple:
Domain of f⁻¹ = Range of f
| Function type |
Range of original function |
Domain of inverse |
| Linear, a ≠ 0 |
All real numbers |
(-∞, ∞) |
| Quadratic, a > 0 |
[vertex y, ∞) |
[vertex y, ∞) |
| Quadratic, a < 0 |
(-∞, vertex y] |
(-∞, vertex y] |
| Rational |
All real y except horizontal asymptote |
Same excluded y value |
| Exponential |
Above or below vertical shift |
Same interval |
| Logarithmic |
All real numbers |
(-∞, ∞) |
| Square root |
Starts at vertical shift |
Same ray interval |
Understanding the domain of an inverse function
Core idea
The domain of an inverse function comes from the range of the original function. This rule is the main shortcut. First study every output that the original function can produce. Then use those outputs as valid inputs for the inverse. This calculator follows that method for common algebraic models.
Why range matters
An inverse reverses the roles of x and y. The original output becomes the inverse input. So a value cannot enter the inverse unless the original function produced it. For a line with nonzero slope, every real output is possible. Its inverse domain is all real numbers. For a quadratic, the vertex controls the lowest or highest output.
One-to-one checks
A true inverse function also needs the original function to be one-to-one. This means no two input values should share the same output. Linear, exponential, logarithmic, and square root models usually pass this test under valid settings. A full quadratic does not pass it. It needs a left or right branch restriction.
Important function families
Exponential functions have a horizontal asymptote. Their ranges stay above or below the vertical shift. Rational functions often exclude the horizontal asymptote from the range. Logarithmic functions can produce every real output. Square root functions begin at a start value and move in one direction only.
Using the graph
The graph helps confirm the interval. Look at the y-values reached by the original curve. Those y-values form the inverse domain. The sample table gives numerical support. It does not replace exact algebra, but it makes the result easier to inspect. Use a wider graph window when behavior is hidden near an asymptote or vertex.
FAQs
1. What is the domain of an inverse function?
It is the set of inputs allowed in the inverse function. It equals the range of the original function because inverse functions swap input and output roles.
2. Why does a quadratic need a branch restriction?
A full quadratic usually fails the horizontal line test. Two x-values can give one y-value. Restricting to the left or right branch makes it one-to-one.
3. Is the inverse domain always all real numbers?
No. It is all real numbers only when the original function range is all real numbers. Lines, cubics, and logarithmic functions often have this result.
4. What does an excluded value mean?
An excluded value is not allowed as an inverse input. It usually appears when the original function never reaches that output, such as a horizontal asymptote.
5. Can this calculator handle rational functions?
Yes. It checks the horizontal asymptote and determinant behavior for linear rational forms. It also warns when the model becomes constant or invalid.
6. How is the exponential inverse domain found?
The calculator checks the vertical shift. A valid exponential range sits above or below that value, depending on the sign of the main coefficient.
7. Why is the graph useful?
The graph shows the original function outputs visually. This helps confirm the range interval, asymptotes, vertex behavior, and starting points.
8. Does the CSV include all graph points?
The CSV focuses on the result summary. The on-page sample table shows selected graph points for checking the calculated inverse domain.