1.4 Composition of Functions

Composition of Functions

Functions can be combined algebraically or through composition to build new, more complex functions. These operations show up constantly in pre-calc and calculus, so getting comfortable with them now pays off. This section covers algebraic combinations, how to compose functions, how to find their domains, and how to decompose them into parts.

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Combining Functions Algebraically

You can combine two functions at each input value using the standard arithmetic operations:

• Sum: $(f + g)(x) = f(x) + g(x)$
• Difference: $(f - g)(x) = f(x) - g(x)$
• Product: $(f \cdot g)(x) = f(x) \cdot g(x)$
• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, defined only where $g(x) \neq 0$

Each of these works "point-wise," meaning you evaluate both functions at the same $x$ and then combine the results. For the quotient, you must exclude any $x$-values where $g(x) = 0$ from the domain.

Construction of Composite Functions

Composition is different from the arithmetic combinations above. Instead of combining two outputs, you feed one function's output into the other function as its input.

The notation $(f \circ g)(x)$ means $f(g(x))$: evaluate $g$ first, then plug that result into $f$.

Here's the process:

1. Start with your input $x$.
2. Evaluate the inner function $g(x)$.
3. Take that output and use it as the input for the outer function $f$.
4. The final result is $f(g(x))$.

Order matters. Composition is not commutative, so $(f \circ g)(x)$ and $(g \circ f)(x)$ are generally not the same.

For example, let $f(x) = 2x + 1$ and $g(x) = x^2$:

• $(f \circ g)(x) = f(g(x)) = f(x^2) = 2x^2 + 1$
• $(g \circ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1$

Those are clearly different functions, which is why you always need to pay attention to which function is inner and which is outer.

Values of Composite Functions

To evaluate a composite function at a specific number, work from the inside out:

1. Plug the given $x$-value into the inner function $g$.
2. Take that numerical result and plug it into the outer function $f$.

Example: Suppose $f(x) = x^2$ and $g(x) = 3x - 2$. Find $(f \circ g)(1)$.

1. Evaluate the inner function: $g(1) = 3(1) - 2 = 1$

2. Plug into the outer function: $f(1) = 1^2 = 1$

3. So $(f \circ g)(1) = 1$

This extends to chains of three or more functions. For $(f \circ g \circ h)(x) = f(g(h(x)))$, you evaluate $h$ first, then $g$, then $f$. Always work from the innermost function outward.

Domain of Composite Functions

Finding the domain of $(f \circ g)(x)$ requires two conditions to be satisfied simultaneously:

1. $x$ must be in the domain of $g$ (so $g(x)$ is defined).
2. $g(x)$ must be in the domain of $f$ (so $f(g(x))$ is defined).

The domain of the composite is the set of all $x$-values that satisfy both conditions.

Example: Let $f(x) = \sqrt{x}$ and $g(x) = x^2 - 4$.

• The domain of $g$ is all real numbers (no restrictions on a polynomial).
• The domain of $f$ requires its input to be $\geq 0$, so you need $g(x) \geq 0$.
• Solve $x^2 - 4 \geq 0$, which factors as $(x-2)(x+2) \geq 0$. This gives $x \leq -2$ or $x \geq 2$.

So the domain of $(f \circ g)(x) = \sqrt{x^2 - 4}$ is $(-\infty, -2] \cup [2, \infty)$.

Components of Composite Functions

Sometimes you're given a composite function and need to break it apart into its inner and outer pieces. This is called decomposition, and it's a skill you'll use frequently in calculus.

To decompose, look for a "nested" structure: something inside something else.

Example: Decompose $(f \circ g)(x) = \sqrt{2x - 1}$.

• The outer operation is the square root, so $f(x) = \sqrt{x}$.
• The expression inside the square root is $2x - 1$, so $g(x) = 2x - 1$.

Check: $f(g(x)) = f(2x - 1) = \sqrt{2x - 1}$. ✓

Decomposition isn't always unique. For instance, $h(x) = (3x + 5)^4$ could be decomposed as $f(x) = x^4$ and $g(x) = 3x + 5$, but you could also choose $f(x) = x^2$ and $g(x) = (3x+5)^2$. The most natural split is usually the one that identifies the simplest outer and inner functions.

Function Notation and Relationships

• Function notation like $f(x)$ names the output of function $f$ when the input is $x$. The letter $f$ names the rule; the parentheses hold the input.
• Substitution is the core mechanic behind composition. Wherever you see $x$ in the outer function's formula, you replace it with the entire expression for the inner function. This is why writing $f(g(x))$ as $f(\text{stuff})$ and then substituting $g(x)$ for "stuff" can help keep things organized.
• Understanding this input-output chain is what makes composition click: every function transforms its input, and composition just stacks those transformations.