Inverse Function Graphing Calculator
This interactive graphing calculator displays the graph of a function \( f(x) \) (blue) and its inverse (red),
along with the reference line \( y = x \).
By definition, if \( (a, f(a)) \) is a point on the graph of \( f \), then
\( (f(a), a) \) is a point on the graph of the inverse.
The inverse of a function may or may not itself be a function.
If \( f \) is a one-to-one function,
then its inverse is also a function. Otherwise, the inverse is a
relation
but not a function.
How to Use the Inverse Function Calculator
Enter a formula for \( f(x) \), for example \( 2x - 1 \), then click
Plot f(x) and Its Inverse.
- Blue graph: \( f(x) \)
- Red graph: inverse of \( f(x) \)
- Black line: \( y = x \)
Hover over the graph to trace coordinates. Use the Plotly toolbar to download the graph as an image.
Interactive Exploration
-
The graphs of \( f \) and its inverse are reflections across the line \( y = x \).
Try functions such as \( x^3 \) or \( x^2 \).
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Enter \( \sqrt{x} \). Is the inverse graph a function?
Find the inverse analytically and compare domains.
-
Enter \( x^2 \). Is the inverse a function? What does this say about one-to-one functions?
-
Enter \( x^3 - 1 \). Determine whether the inverse is a function and find its formula.
Exercises
Graph each function and its inverse. Decide whether the inverse is a function and explain why.
- \( f(x) = x^4 \)
- \( f(x) = \sin(x) \)
- \( f(x) = \ln(x) \)
- \( f(x) = e^x \)
- \( f(x) = 5 \)
- \( f(x) = \arcsin(x) \)
Solutions
- \( f(x) = x^4 \): inverse exists but is not a function (not one-to-one).
- \( f(x) = \sin(x) \): inverse is not a function.
- \( f(x) = \ln(x) \): inverse is \( f^{-1}(x) = e^x \).
- \( f(x) = e^x \): inverse is \( f^{-1}(x) = \ln(x) \).
- \( f(x) = 5 \): inverse is not a function.
- \( f(x) = \arcsin(x) \): inverse is \( \sin(x) \) with domain \( -\frac{\pi}{2} \le x \le \frac{\pi}{2} \).
Further Reading