📈College Algebra Unit 3 Review

Function composition lets you chain two functions together so that the output of one becomes the input of the next. This shows up constantly in later math courses and in real-world modeling, where multi-step processes naturally feed into each other. Before getting to composition, though, you need to be comfortable combining functions with basic arithmetic.

Combining Functions Algebraically

You can add, subtract, multiply, or divide two functions $f$ and $g$ by performing that operation on their outputs for the same input $x$ .

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: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ , where $g(x) \neq 0$

For the sum, difference, and product, the domain is the intersection of the domains of $f$ and $g$ . Both functions have to be defined at a given $x$ for the combined function to work there.

For the quotient, you take that same intersection but also exclude any $x$ -values where $g(x) = 0$ , since division by zero is undefined.

Example: If $f(x) = \sqrt{x}$ (domain: $x \geq 0$ ) and $g(x) = x - 3$ (domain: all reals), then the domain of $(f + g)$ is $x \geq 0$ . The domain of $\left(\frac{f}{g}\right)$ is $x \geq 0$ with $x \neq 3$ .

Creation of Composite Functions

Composition creates a new function by feeding the output of one function into another:

$(f \circ g)(x) = f(g(x))$

Read this as " $f$ composed with $g$ of $x$ ." The key idea: evaluate the inner function $g$ first, then plug that result into the outer function $f$ .

To evaluate a composite function $(f \circ g)(a)$ at a specific value:

Calculate $g(a)$ .
Take that result and plug it into $f$ . Your answer is $f(g(a))$ .

Example: If $f(x) = x^2$ and $g(x) = x + 3$ , then $(f \circ g)(2) = f(g(2)) = f(5) = 25$ .

Order matters. Composition is not commutative. Using the same functions above, $(g \circ f)(2) = g(f(2)) = g(4) = 7$ , which is a completely different answer. Always pay close attention to which function is inner and which is outer.

Combining functions algebraically, Find domains and ranges of the toolkit functions | College Algebra

Domain of Composite Functions

Finding the domain of $(f \circ g)$ requires a two-part check:

Start with the domain of the inner function $g$ . The input $x$ must be in $g$ 's domain.
Check that $g(x)$ lands in the domain of $f$ . The output of $g$ has to be a valid input for $f$ .
The domain of $(f \circ g)$ is the set of all $x$ -values that satisfy both conditions.

Example: Let $f(x) = \sqrt{x}$ and $g(x) = 4 - x^2$ . The domain of $g$ is all reals. But $f$ requires its input to be $\geq 0$ , so you need $g(x) \geq 0$ , meaning $4 - x^2 \geq 0$ , which gives $-2 \leq x \leq 2$ . That's the domain of $(f \circ g)$ .

Components of Composite Functions

Sometimes you'll be given a composite function and asked to break it into its component parts. The strategy is to identify what happens "inside" versus "outside."

The inner function $g$ is applied first (it's the part closest to $x$ ).
The outer function $f$ is applied last (it's the operation that wraps around everything).

Example: If $h(x) = (2x + 1)^3$ , you could decompose this as $f(x) = x^3$ and $g(x) = 2x + 1$ , so $h(x) = f(g(x))$ . There's sometimes more than one valid decomposition, but look for the most natural split.

Combining functions algebraically, Find domain and range from graphs | College Algebra

Applications of Function Composition

Composition models any situation where one process feeds into another:

Unit conversion: If $f$ converts inches to feet and $g$ converts feet to meters, then $(g \circ f)$ converts inches directly to meters.
Course grades: If $f$ converts raw scores to percentages and $g$ converts percentages to letter grades, then $(g \circ f)$ takes raw scores straight to letter grades.
Business calculations: If $R(x)$ gives revenue from $x$ units sold and $C(x)$ gives cost, you could compose these with other functions to model multi-step financial processes.

The pattern is always the same: identify which process happens first (that's your inner function) and which happens second (that's your outer function).

Special Functions and Relationships

A one-to-one function produces a unique output for every unique input. Only one-to-one functions have inverse functions.
An inverse function "undoes" the original. If $f(a) = b$ , then $f^{-1}(b) = a$ . When you compose a function with its inverse, you get back to where you started: $f(f^{-1}(x)) = x$ .
The identity function $f(x) = x$ leaves every input unchanged. It plays a role similar to multiplying by 1 or adding 0: composing any function with the identity function returns the original function.

📈College Algebra Unit 3 Review