Functions are like building blocks that can be combined in various ways. We can add, subtract, multiply, and divide them, creating new functions with unique properties. This algebraic manipulation allows us to model complex relationships and solve real-world problems.

Composition is a special way of combining functions, where one function's output becomes another's input. This nested structure creates new functions with interesting behaviors. Understanding composition helps us break down complex functions and analyze their properties.

Composition of Functions

Combining functions algebraically

• Add functions together point-wise $(f + g)(x) = f(x) + g(x)$ (sum of $f$ and $g$ at each input value $x$)
• Subtract functions point-wise $(f - g)(x) = f(x) - g(x)$ (difference between $f$ and $g$ at each input value $x$)
• Multiply functions point-wise $(f \cdot g)(x) = f(x) \cdot g(x)$ (product of $f$ and $g$ at each input value $x$)
• Divide functions point-wise $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (quotient of $f$ and $g$ at each input value $x$), where $g(x) \neq 0$ to avoid division by zero

Construction of composite functions

• Construct a composite function $(f \circ g)(x)$ by substituting the output of the inner function $g(x)$ as the input of the outer function $f$
• Evaluate the inner function $g(x)$ first, then use the result as the input for the outer function $f$ to obtain $f(g(x))$
• Order matters in composition $(f \circ g)(x) = f(g(x))$ is not always equal to $(g \circ f)(x) = g(f(x))$ (composition is not commutative)
• If $f(x) = 2x + 1$ and $g(x) = x^2$, then $(f \circ g)(x) = f(g(x)) = 2x^2 + 1$ and $(g \circ f)(x) = g(f(x)) = (2x + 1)^2$
• Nested functions are a common way to represent composite functions, where one function is contained within another (e.g., $f(g(x))$)

Values of composite functions

• Calculate the value of a composite function $(f \circ g)(x)$ at a specific input $x$ by first evaluating the inner function $g(x)$, then using the result as the input for the outer function $f$
• If $(f \circ g)(x) = (3x - 2)^2$ and $x = 1$, first evaluate $g(1) = 3(1) - 2 = 1$, then evaluate $f(1) = 1^2 = 1$, so $(f \circ g)(1) = 1$
• Composition can be extended to more than two functions $(f \circ g \circ h)(x) = f(g(h(x)))$ (evaluate from innermost to outermost)
• Function evaluation involves determining the output of a function for a given input, which is crucial in working with composite functions

Domain of composite functions

• The domain of a composite function $(f \circ g)(x)$ consists of all values of $x$ that are in the domain of $g$ and for which $g(x)$ is in the domain of $f$
• If $f(x) = \sqrt{x}$ (domain: $x \geq 0$) and $g(x) = x^2 - 4$ (domain: all real numbers), then for $(f \circ g)(x)$ to be defined, $g(x) \geq 0$, so $x^2 - 4 \geq 0$, which gives $x \leq -2$ or $x \geq 2$
• The domain of $(f \circ g)(x)$ is the intersection of the domain of $g$ and the set of $x$ values for which $g(x)$ is in the domain of $f$

Components of composite functions

• Identify the outer function $f$ and inner function $g$ in a composite function $(f \circ g)(x)$
• The inner function $g$ is evaluated first, and its output becomes the input of the outer function $f$
• In the composite function $(f \circ g)(x) = \sqrt{2x - 1}$, the outer function is $f(x) = \sqrt{x}$ and the inner function is $g(x) = 2x - 1$
• Decomposing a composite function into its component functions helps in understanding its structure and behavior

Function Notation and Relationships

• Function notation (e.g., $f(x)$) is used to represent the output of a function for a given input $x$
• The input-output relationship in functions describes how the input values are mapped to output values
• Substitution is the process of replacing a variable in a function with a specific value or expression