Inverse Functions

Understanding Inverse Functions

An inverse function reverses the correspondence established by an original function, denoted as f⁻¹(x) for the original function f(x). Inverse functions exist exclusively for bijective functions, which are both one-to-one (injective) and onto (surjective), ensuring that each element of the domain maps to a unique element in the range and every element in the range is mapped from the domain. To find an inverse, one typically replaces f(x) with y, rearranges the equation to solve for x in terms of y, and then interchanges x with f⁻¹(x) and y with x to express the inverse function.

Procedure for Finding Inverse Functions

To illustrate the process of finding an inverse function, consider the function f(x) = 5x + 6. Start by replacing f(x) with y, resulting in y = 5x + 6. Then, solve for x by rearranging the equation to x = (y - 6) / 5. Finally, interchange y with x and x with f⁻¹(x) to obtain the inverse function f⁻¹(x) = (x - 6) / 5. For the function j(x) = x² - 6, which is not one-to-one over all real numbers, we must restrict the domain to non-negative numbers for the function to have an inverse. Starting with y = x² - 6 and solving for x gives x = ±√(y + 6). By restricting the domain to x ≥ 0, the inverse function is j⁻¹(x) = √(x + 6).

Solving Inverse Function Problems

Inverse function problems can vary in complexity. If given f⁻¹(x) and asked to find its value for a specific x, simply substitute the given x value into the inverse function. For example, to evaluate f⁻¹(4) when f⁻¹(x) = 6x - 2, replace x with 4 to calculate f⁻¹(4) = 6(4) - 2, which equals 22. Another type of problem involves finding x when the inverse function is set equal to a known value. If g⁻¹(x) = 6x + 4 and g⁻¹(x) is given as 58, set 58 = 6x + 4 and solve for x to find x = 9.

Domains and Ranges of Inverse Functions

The domain of an original function becomes the range of its inverse, and the range of the original function becomes the domain of the inverse. For the function h(x) = 3x² + 4, which is not one-to-one over all real numbers, we must restrict the domain to x ≥ 5 to ensure it has an inverse. The inverse function is then h⁻¹(x) = √((x - 4) / 3), and its domain is the range of h(x) for x ≥ 5. To find the range of h⁻¹(x), we consider the domain of h(x), which yields h(5) = 79, indicating that the range of h⁻¹(x) is x ≥ 79.

Graphical Representation of Inverse Functions

Graphically, an inverse function is the reflection of the original function across the line y = x. To visualize this, one can reflect the graph of the original function across this line. Alternatively, after determining the inverse function algebraically, plot its points on a graph. For example, to graph the inverse function g⁻¹(x) = (x - 2)² - 4, where the domain is restricted to 0 ≤ x ≤ 6 to ensure it is one-to-one, calculate the corresponding y values, plot the points, and connect them to represent the inverse function.

Key Takeaways on Inverse Functions

Inverse functions are mathematical counterparts to original functions, indicated by a superscript -1. They are defined for bijective functions and are found by interchanging the roles of x and y in the original function and solving for the new y. It is essential to understand the interdependent relationship between the domains and ranges of functions and their inverses. Graphical representations of inverse functions can be achieved by reflecting the original function across the line y = x or by plotting the inverse function's points directly on a graph.