Algebra - Inverse Functions

Inverse Functions

An inverse function is a function that inverts another function back to the original function. A function is notated f(x) – stated f of x, (f(x) = y). An inverse function is notated as f -1(x) – stated function inverse of x, or inverse function of x. Do not mistake the superscript of (-1) as an exponent. The -1 is a symbol for the inverse function of x.

Vertical Line Test

For a set of numbers to be a function, it must pass the vertical line test. A function is defined as, the relation of x and y values; in which one x value is associated with exactly one y value. The vertical line test is a graphical representation to verify if a set of numbers are a function. In the vertical line test, a vertical line passes through no more than one value of x for one y value. Therefore, two different x values can lead to the same y value. For example, an x2 parabola (Figure 1, x2 parabola) (squarecirclez.com, 2010) can be a function, a y2 parabola (Figure 2, y2 parabola) (squarecirclez.com, 2010) cannot.

(Figure 1, x2 parabola) (Figure 2, y2 parabola)

Horizontal Line and One to One

Not all functions have an inverse function. An inverse function must satisfy the vertical line test as well as the horizontal line test. In the horizontal line test, the function can only intersect the horizontal line in at most, one place. An inverse function must also be a one to one function: every x value can only have one y value, and the y value came from only one x value. The one to one function also satisfies the horizontal line test.

The f(x) and f -1(x) Process

1. The first example is a function: f(x) = y.

y = 2x + 8, find f(5).

y = 2(5) + 8

= 10 + 8

y = 18

A function takes the input x, which in this case is 5,...