Combining Functions: Arithmetic Operations

Ever wondered how mathematicians build complex functions from simpler ones? It's like creating a recipe by combining ingredients! When we have two functions f and g, we can combine them in several ways.

The four basic operations create new functions with specific domains:

• Sum: f+g$$x = f$x$ + g$x$
• Difference: f-g$$x = f$x$ - g$x$
• Product: (fg)$x$ = f$x$ × g$x$
• Quotient: (f/g)$x$ = f$x$ ÷ g$x$, where g$x$ ≠ 0

For example, if f$x$ = 3x+1 and g$x$ = 2x²-1, we can find:

• g-f$$x = 2x² - $3x+1$ = 2x² - 3x - 2
• f+g$$x = $3x+1$ + $2x²-1$ = 2x² + 3x
• (g/f)$x$ = $2x²-1$ ÷ $3x+1$, which is defined when x ≠ -⅓

Pro Tip: When finding domains of combined functions, remember that both functions need to be defined at that input value. For quotients, you also need to check where the denominator equals zero!

Function Composition

Function composition is like a mathematical assembly line! When we write (f ∘ g)$x$ = f(g$x$), we're saying "take input x, run it through function g, then use that result as input for function f."

The domain of f ∘ g includes all values of x where:

1. x is in the domain of g
2. g$x$ is in the domain of f

Let's practice with f$x$ = 3x+1 and g$x$ = 2x²-1:

• (f ∘ g)$x$ = f(g$x$) = f$2x²-1$ = 3$2x²-1$+1 = 6x²-2
• (g ∘ f)$x$ = g(f$x$) = g$3x+1$ = 2$3x+1$²-1 = 18x²+12x+1

Notice that (f ∘ g)$x$ ≠ (g ∘ f)$x$ in most cases. Order matters in function composition!

Remember: Function composition isn't commutative! Think of it like putting on socks and shoes—you can't switch the order and get the same result.

Advanced Composition and Decomposition

You can compose more than just two functions! For three functions, we follow the pattern (f ∘ g ∘ h)$x$ = f(g(h$x$)). This means we apply h first, then g, and finally f.

For example, if f$x$ = x/$x+1$, g$x$ = x¹⁰, and h$x$ = x+3, then:

• (f ∘ g ∘ h)$x$ = f(g(h$x$)) = f$g(x+3)$ = f$(x+3)¹⁰$ = $x+3$¹⁰/$(x+3)¹⁰+1$

Function decomposition is like working backward—finding simpler functions that compose to make a more complex one. For F$x$ = √$x+9$, we can decompose it into:

• g$x$ = x+9 (the inner function)
• f$x$ = √x (the outer function)
• (f ∘ g)$x$ = f(g$x$) = f$x+9$ = √$x+9$ = F$x$

Challenge yourself: When you see a complex function, try to identify if it could be written as a composition of simpler functions. This skill helps solve many calculus problems later!