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.