Inverse Function Graphing Calculator

An online graphing calculator to draw the graph of function f (in blue) and its inverse (in red). The graph of the inverse of f is fomed by reversing the ordered pairs corresponding to all points on the graph (blue) of a function f. If (a , f(a)) is a point on the graph of f then the point (f(a) , a) is a point on the graph of the inverse of f. The inverse of a function may or may not be a function. If function f is a one-to-one function, the graph of the inverse is that of a function. If function f is not a one to one, the inverse is a relation but not a function. If needed, Free graph paper is available.

How to Use Inverse Functions Graphing Calculator

Enter a formula for function f (2x - 1 for example) and press "Plot f(x) and Its Inverse". Three graphs are displayed: the graph of function f (blue) that you input, the line y = x (black), and the graph (red) of the inverse.
The variable in the expression of the function is the small letter x.

f(x) =

Hover the mousse cursor over the graph to trace the coordinates.
Hover the mousse cursor on the top right of the graph to have the option of download the graph as a png file.

Interactive Tutorial

  1. Due to the definition of the inverse function, we can easily note that the graphs of f (blue) and its inverse (red) are reflection of each other on the line y = x. Enter different functions (x^3 , x^2, ...)and explore this property of reflection of the graphs of f and its inverse.
  2. Enter sqrt(x) in the editing window of the calculator, which means f(x) = sqrt(x), and press "Plot f(x) and Its Inverse". sqrt stands for square root. Is the graph of the reversed pairs (in red) that of a function? If it is the graph of a function, it is the inverse function of f(x) = sqrt(x). Find, analytically, the inverse function of f(x) = sqrt(x) including the domain and compare it to the graph obtained.
  3. Enter x^2 in the editing window, which means f(x) = x^2, and press "Plot f(x) and Its Inverse". Is the graph of of the inverse (in red) that of a function? If it is not a function, is f a one to one function?
  4. Enter x^3 - 1 in the editing window, which means f(x) = x^3 - 1, and press enter. Is the graph of of the inverse (in red) that of a function? If it is, it has to be the graph of the inverse of f. Find a formula for the graph of the inverse of f.

Exercises

Use the graphing calculator above to graph each of the functions below and its inverse and determine whether the inverse (red) is a function or not and give an explanation.
  1. f(x) = x 4 (type as x^4)
  2. f(x) = sin(x)
  3. f(x) = ln(x) ( type as log(x))
  4. f(x) = e^x
  5. f(x) = 5
  6. f(x) = arcsin(x) ( type as asin(x)).

Solutions to Above Exercies

  1. f(x) = x^4 has an inverse but it is not a function becaues f it is not a one to one.
  2. f(x) = sin(x) : The inverse of f(x) = sin(x) is not a function becaues f it is not a one to one
  3. f(x) = ln(x) : The inverse f(x) = ln(x) is a function given by ex
  4. f(x) = exp(x) : The inverse f(x) = exp(x) is a function given by ln(x)
  5. f(x) = 5 : The inverse of f(x) = 5 is not a function becaues f it is not a one to one
  6. f(x) = arcsin(x) : The inverse of f(x) = arcsin(x) is a function given by sin(x) with restricted domain -pi/2 > x > pi/2

More References and Links

Inverse Function.
One-To-One Functions.
Find The Inverse of a Relation - Questions With Detailed Solutions.
Graphing Calculators.

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