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 2*x - 1, then click Plot f(x) and Its Inverse.

\( f(x) = \)

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

Interactive Exploration

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

Exercises & Solutions

Practice Exercises & Detailed Solutions

Graph each function and its inverse. Decide whether the inverse is a function and explain why.

  1. \( f(x) = x^4 \)
    Solution: Inverse exists algebraically as \( x = \pm\sqrt[4]{y} \), but is not a function (fails the vertical line test; not one-to-one).
  2. \( f(x) = \sin(x) \)
    Solution: Inverse (\(\arcsin(x)\)) is a relation, not a function across all real numbers unless restricted to a principal domain.
  3. \( f(x) = \ln(x) \)
    Solution: Inverse is \( f^{-1}(x) = e^x \), which is a function.
  4. \( f(x) = e^x \)
    Solution: Inverse is \( f^{-1}(x) = \ln(x) \), which is a function.
  5. \( f(x) = 5 \)
    Solution: Constant function; inverse is a horizontal line \( x = 5 \), which is not a function.
  6. \( f(x) = \arcsin(x) \)
    Solution: Inverse is \( \sin(x) \) restricted to domain \( -\frac{\pi}{2} \le x \le \frac{\pi}{2} \).

Further Reading & Resources

Explore additional tutorials and tools:

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