Inverse Function Graphing Calculator - Free Mathematics Tutorials

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: \( y = x \)

\( f(x) = \) 2*x - 1

Hover over the graph to trace coordinates. Use the Plotly toolbar to download the graph as an image.

Interactive Exploration

1. The graphs of \( f \) and its inverse are reflections across the line \( y = x \). Try functions such as \( x^3 \) or \( x^2 \).
2. Enter \( \sqrt{x} \). Is the inverse graph a function? Find the inverse analytically and compare domains.
3. Enter \( x^2 \). Is the inverse a function? What does this say about one-to-one functions?
4. 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.

1. \( f(x) = x^4 \)
2. \( f(x) = \sin(x) \)
3. \( f(x) = \ln(x) \)
4. \( f(x) = e^x \)
5. \( f(x) = 5 \)
6. \( f(x) = \arcsin(x) \)

Solutions

1. \( f(x) = x^4 \): inverse exists but is not a function (not one-to-one).
2. \( f(x) = \sin(x) \): inverse is not a function.
3. \( f(x) = \ln(x) \): inverse is \( f^{-1}(x) = e^x \).
4. \( f(x) = e^x \): inverse is \( f^{-1}(x) = \ln(x) \).
5. \( f(x) = 5 \): inverse is not a function.
6. \( f(x) = \arcsin(x) \): inverse is \( \sin(x) \) with domain \( -\frac{\pi}{2} \le x \le \frac{\pi}{2} \).

Further Reading

• Inverse Functions
• One-To-One Functions
• Inverse of a Relation
• More Graphing Calculators