The Dynamics of Composite Functions

What Are Composite Functions?

A composite function is like using two function machines in a row. You take the output of one function and feed it into another function.

For example:

• Function 1: $f(x) = x + 2$ (Add 2 to $x$)
• Function 2: $g(x) = 3x$ (Multiply $x$ by $3$)
• To find $g(f(x))$ or $gf(x)$: First apply $f(x)$, then apply $g(x)$ to the result

We write composite functions like this:

• $g(f(x))$ or $gf(x)$: Read as "$g$ of $f(x)$".
• $f(g(x))$ or $fg(x)$: Read as "$f$ of $g(x)$".

 

How to Solve Composite Functions

1. Start with the inside function (e.g., $f(x)$)
2. Calculate its result using the given $x$
3. Use that result as the input for the outside function (e.g., $g(x)$)

 

Examples

Example 1: Find $gf(x)$ for $f(x) = x + 2$ and $g(x) = 3x$:

Write out $gf(x)$: $$gf(x) = g(x + 2)$$

Substitute $x + 2$ into $g(x)$: $$g(x + 2) = 3(x + 2)$$

Expand: $$gf(x) = 3x + 6$$

 

 

 

 

 

 

Example 2: Evaluate $fg(x)$ for the same functions when $x = 4$:

Write out $fg(x)$: $$fg(x) = f(3x)$$

Substitute $3x$ into $f(x)$: $$f(3x) = 3x + 2$$

Evaluate at $x = 4$: $$fg(4) = 3(4) + 2 = 12 + 2 = 14$$

 

 

 

 

 

Example 3: Find $h(g(f(x)))$ for $f(x) = x + 1, g(x) = x^2$, and $h(x) = 2x - 3$:

Start with $f(x)$ : $$f(x) = x + 1$$

Substitute $f(x)$ into $g(x)$: $$gf(x) = g(x + 1) = (x + 1)^2$$

Substitute $gf(x)$ into $h(x)$: $$h(g(f(x))) = h((x + 1)^2) = 2(x + 1)^2 - 3$$

Expand $(x + 1)^2$: $$h(g(f(x))) = 2(x^2 + 2x + 1) - 3 = 2x^2 + 4x + 2 - 3$$

Simplify: $$h(g(f(x))) = 2x^2 + 4x - 1$$