How are a function's graph and its inverse related?
Quick answer
A function and its inverse reverse input and output: if f(a) = b, then f⁻¹(b) = a. So each point (a, b) on the function becomes (b, a) on its inverse, reflecting the graph across y = x. This works when the function is one-to-one on its domain. For example, f(x) = x² with x ≥ 0 has inverse f⁻¹(x) = √x.
The question, the details, the next step.
How should I picture the relationship between a function's graph and its inverse? I know the rule about reversing inputs and outputs, but I’m not sure what that means visually.
Picture the line y = x as a mirror. For a one-to-one function, every input-output pair reverses in its inverse: if f(a) = b, then f⁻¹(b) = a. That changes the graph point (a, b) into (b, a), so the inverse graph is the reflection of the original across y = x. [S1]
For example, if a function contains the point (2, 4), its inverse contains (4, 2). The point moves to the other side of the mirror line; it does not keep the same coordinates. A point on y = x, such as (3, 3), stays in the same place when reflected.
Can you show the coordinate swap with an actual function and its inverse?
Take f(x) = x², but use the domain x ≥ 0. On that domain its inverse is f⁻¹(x) = √x. [S1]
Here are paired points:
- On f, x = 0 gives y = 0, so (0, 0) maps to (0, 0) on the inverse.
- On f, x = 1 gives y = 1, so (1, 1) maps to (1, 1).
- On f, x = 2 gives y = 4, so (2, 4) maps to (4, 2).
- On f, x = 3 gives y = 9, so (3, 9) maps to (9, 3).
Each inverse point has the original point’s coordinates reversed. To sketch both graphs, plot these pairs and their swapped partners, then use y = x as the reflection line.
Why did you restrict the square function to x ≥ 0? Wouldn’t the same coordinate-swapping idea work for all x?
The coordinate swap still describes reversed pairs, but an inverse must itself be a function. On the unrestricted square function, both x = 2 and x = −2 produce y = 4. Reversing those pairs would give the input 4 two outputs, 2 and −2, so the inverse relation would not be a function. The square function is not one-to-one on its full domain; restricting it to x ≥ 0 makes the inverse function √x. [S1]
When working with another function, check whether distinct allowed inputs ever give the same output. If they do, the inverse relation may not be a function unless the original domain is restricted. Whether a particular function is one-to-one depends on that function and its domain.
What’s a reliable way to check my own sketch or a table of values?
Use the original points as a checklist: if (a, b) is on f, verify that (b, a) is on the inverse. For the example, (2, 4) on f(x) = x² corresponds to (4, 2) on f⁻¹(x) = √x, because √4 = 2. A common slip is to change the sign or only swap part of the point; reverse the full coordinate order.
You can also check the picture: corresponding points should be equally placed on opposite sides of y = x, and points on that line remain fixed. These checks do not replace checking that the stated inverse matches the function’s domain.
A practical checklist
Write down the original function’s allowed domain.
Check whether different allowed inputs can produce the same output.
For each point (a, b), make the inverse point (b, a).
Plot y = x as a reference mirror and compare paired points.
Check the inverse rule by substituting swapped input-output values.
A few more useful answers
Does every function have an inverse function?
Not necessarily. The function must be one-to-one on its domain for its inverse relation to be a function. A domain restriction can sometimes make that possible, as with x² restricted to x ≥ 0. [S1]
Do the graphs of a function and its inverse always cross on y = x?
The reflection relationship means points on y = x stay fixed, but it does not by itself guarantee that a particular function and its inverse have a point in common.
When I reflect a point, do I change its signs?
No. Reflection across y = x swaps the coordinates: (a, b) becomes (b, a). For example, (2, 4) becomes (4, 2), not (−2, −4).
Sources
References used for this guide.
- [S1] OpenStax College Algebra: Inverse Functions openstax.org · Checked