When we learn about functions, it's really important to know the differences between function composition and function addition. Let’s make it simple!

Function Composition

• What It Is: This is when you take two functions and make a new one by using one function on the result of another.
• How We Write It: If we have two functions, $f(x)$ and $g(x)$, we write the composition as $(f \circ g)(x) = f(g(x))$.
• An Example: Let’s say $f(x) = 2x$ and $g(x) = x + 3$.
  • If we find $(f \circ g)(x)$, we get:
  • First, find $g(x)$: $g(x) = x + 3$.
  • Now put that into $f$: $f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6$!

Function Addition

• What It Is: This is when you add the results of two functions together for the same input $x$.
• How We Write It: We write this as $(f + g)(x) = f(x) + g(x)$.
• An Example: Using our same functions, we can add them:
  • $f(x) + g(x) = 2x + (x + 3) = 3x + 3$!

To Sum It Up

• In composition, we take outputs and use them as new inputs.
• In addition, we just add the outputs together.

What a fun way to explore functions!