Composite and Inverse functions

Composite and Inverse functions

Composite Functions

Understanding Composite Functions

  • A composite function is a function that is created from two given functions by using the output from the first function as the input for the second function.
  • It is often represented as f(g(x)) or (f o g)(x), where f(x) and g(x) are two different functions.

Forming Composite Functions

  • When forming the composite function f(g(x)), you substitute the whole function g(x) into the function f(x) wherever x appears.
  • For example, if f(x) = x² + 3 and g(x) = 2x + 1, the composite function f(g(x)) will be (2x+1)² + 3.

Evaluating Composite Functions

  • To evaluate a composite function at a specific value, first evaluate the inner function at that value and then substitute this result into the outer function.

Inverse Functions

Understanding Inverse Functions

  • An inverse function is a function that “reverses” the effect of the original function. If the function f takes x to y, then its inverse takes y back to x.
  • The inverse function of f is usually denoted as f^-1 (read as “f inverse”).

Finding Inverse Functions

  • In order to find the inverse of a given function, swap x and y in the equation of the function, then solve this new equation for y.
  • For example, to find the inverse of the function f(x) = 2x + 3, we swap x and y to get x = 2y + 3. Solving this equation for y gives us y = (x - 3)/2, so f^-1(x) = (x - 3)/2.

Evaluating Inverse Functions

  • To evaluate an inverse function at a specific value, replace ‘x’ in the inverse function’s formula with the given value and solve the equation.
  • An important point about the function and its inverse is that f(f^-1(x)) = x and f^-1(f(x)) = x.

Properties of Inverse Functions

  • If a function has an inverse, the original function and its inverse are reflections of each other in the line y=x when graphed.

Practice both exercises of creating composite and inverse functions from given functions, and of evaluating them at specific points. Understand the relations between a function and its inverse for a complete understanding.