Inverse functions

Understanding Inverse Functions

An inverse function is the opposite of the original function, represented by f^-1(x), while the original function is f(x).

Inverse functions exist only for one-to-one functions, where one input value leads to a single output value. This is unlike one-to-many functions, where one input value can yield multiple output values.

How to Find the Inverse of a Function

There are three simple steps to finding the inverse of a function:

• Substitute y for the function notation (e.g. f(x) becomes y).
• Rearrange the function to make x the subject.
• Replace x with the inverse function notation and y with x (e.g. x becomes f^-1(x)).

For example, to find the inverse function of f(x) = 3x - 2:

• Substitute y: y = 3x - 2.
• Rearrange to make x the subject: x = (y+2)/3.
• Replace x and y with inverse function notation: f^-1(x) = (x+2)/3.

Solving Inverse Function Problems

Inverse functions can be used to solve different types of questions:

• When x is given: substitute the value of x into the inverse function and solve.
• When the function is set to a value: set the inverse function equal to y, rearrange the equation to isolate x, and solve for x.
• For domains and ranges: the domain of the inverse function is the range of the original function, and vice versa.

Graphical Representation of Inverse Functions

There are two ways to graphically represent inverse functions:

1. Directly reflect the original function in the line y = x, using transformation skills for graphs.
2. Find the inverse function and plot the coordinates of x and y on a graph.

To directly reflect the original function in the line y = x, use the original function and the line as the line of reflection. For instance, f(x) = 2x + 5 can be graphed as:

• The original function (red).
• The original function (red) and line y = x (blue) as the line of reflection.
• The inverse function (green) obtained by reflecting the original function (red) in the line of reflection (blue).

If the original function involves a variable raised to a power other than 1, the process may be more complicated.

Inverse Functions - Key Points

• Inverse functions are the opposite of the original function.
• Their notation differs from typical functions, using f^-1(x).
• Only one-to-one functions have inverse functions.
• Inverse functions can be found by substituting y for the function notation, rearranging the equation, and replacing variables with inverse function notation.
• The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.