How to Write f of g?

"f of g" refers to the composition of two functions, denoted as (f ∘ g)(x), which is equivalent to f(g(x)).

Here's a breakdown of how to write and understand function composition:

Understanding Function Composition

The expression "f of g of x," or (f ∘ g)(x), means you first apply the function g to the input x, and then you apply the function f to the result. In other words, the output of g(x) becomes the input for f(x).

How to Write f of g

The most common ways to write "f of g" are:

• (f ∘ g)(x): This is the formal notation, using the composition operator "∘". It's read as "f composed with g of x" or "f of g of x".
• f(g(x)): This is the functional notation, which is often more intuitive. It directly shows that the output of g(x) is used as the input for f(x).

Example

Let's say you have two functions:

• f(x) = x²
• g(x) = x + 1

Then, "f of g of x" would be written as:

• (f ∘ g)(x) = f(g(x))

To find the explicit expression, you substitute g(x) into f(x):

• f(g(x)) = f(x + 1) = (x + 1)² = x² + 2x + 1

Therefore, (f ∘ g)(x) = x² + 2x + 1

Steps to Calculate f(g(x))

1. Identify the functions: Determine the formulas for f(x) and g(x).
2. Substitute: Replace every instance of 'x' in f(x) with the entire expression for g(x).
3. Simplify: Simplify the resulting expression to obtain the composite function.

In Summary

"f of g" represents the composition of functions, where the output of the inner function, g(x), becomes the input of the outer function, f(x). You can write it as (f ∘ g)(x) or f(g(x)), and you calculate it by substituting the expression for g(x) into f(x) and simplifying.