College algebra inverse functions

Overview of Inverse Functions

  • The domain of a function becomes the range of its inverse function.

  • The range of a function corresponds to the domain of its inverse.

  • To find the inverse function, swap the x and y values of ordered pairs.

Definitions

  • Function (f): A relation where each input (x) has a single output (y).

  • Inverse Function (f^(-1)): A function that reverses the effect of the original function; it takes an output value and returns the original input value.

  • One-to-One Function: A function where each output is associated with exactly one input, necessary for an inverse to exist.

Finding the Inverse

  • For ordered pairs: (x, y) becomes (y, x).

  • Example Ordered Pair: For f(1) = 0, the inverse f^(-1)(0) = 1.

  • Not all functions have inverses; only one-to-one functions qualify.

Graphing Inverse Functions

  • To graph an inverse: identify points from the original function, swap x and y, and plot new points.

  • A function passes the vertical line test if any vertical line crosses it only once.

  • A function passes the horizontal line test if any horizontal line crosses it only once, indicating it's one-to-one.

Example Process of Finding Inverse

  1. Start with the function in the form y = f(x).

  2. Replace y with x and vice versa to find the inverse.

  3. Solve for y in terms of x to express the inverse function correctly.

Step-By-Step Example

  • Given f(x) = 2y/(y-3), swap x and y:

    • x = 2y/(y - 3)

  • Multiply both sides by (y - 3) to eliminate fractions:

    • x(y - 3) = 2y

  • Expand and isolate y:

    • xy - 3x = 2y

    • Rearranging gives: xy - 2y = 3x

    • Factor: y(x - 2) = 3x

  • Solve for y:

    • y = 3x/(x - 2)

  • Thus, the inverse function is f^(-1)(x) = 3x/(x - 2).

Verification of Inverses

  • To check if two functions are inverses: verify that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x holds true.

  • This confirms that each function undoes the other.