Summary
Highlights
A function has an input (x, independent variable, domain) and an output (y, dependent variable, range). Inverse functions reverse this process, interchanging the input and output. Examples include multiplying then dividing by two, or squaring then taking the square root. For a data set, finding the inverse means swapping the x and y values.
For an equation like y = 2x - 1, the inverse can be found intuitively by reversing the operations: add 1, then divide by 2, resulting in y = (x+1)/2. Algebraically, you interchange x and y (x = 2y - 1) and solve for the new y. This algebraic method is more formal.
Graphically, a function and its inverse are reflections of each other over the line y = x. This reflection mirrors the interchange of x and y coordinates.
When dealing with functions like f(x) = 1/3x + 7, convert f(x) to y, interchange x and y, and solve for y. The inverse notation is f⁻¹(x). For more complex functions with x in multiple places, such as f(x) = (2x + 3) / (x - 4), interchange x and y, cross-multiply, gather terms with y on one side, factor out y, and then solve for y.
The vertical line test (VLT) determines if a graph is a function. The horizontal line test (HLT) determines if the inverse of a graph is a function. If a function passes the HLT, its inverse will be a function. Failing the HLT means the inverse is not a function.
To verify if two functions, f(x) and g(x), are inverses, compose them both ways: find f(g(x)) and g(f(x)). If both compositions result in x, then the functions are inverses of each other. This demonstrates that one function 'undoes' the other.
Sometimes the inverse of a function, such as a parabola, is not a function itself because it fails the horizontal line test. To make the inverse a function, we must restrict the domain of the original function. For example, by considering only half of a parabola (e.g., x ≥ 0), the inverse will pass the vertical line test. Also, the domain and range swap between a function and its inverse.