Precalculus Study Guide: Functions, Domains, Ranges, and Composition

Functions: Domain, Range, and Evaluation

Understanding Functions

A function is a relation that assigns exactly one output value for each input value from its domain. Functions are fundamental in precalculus and are used to model relationships between quantities.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) that the function can produce.

Evaluating Functions

To evaluate a function at a specific value, substitute the input value into the function's formula or use a table/graph if provided.

• Example: If $f(x) = x^2 + 1$, then $f(2) = 2^2 + 1 = 5$.

Piecewise Functions

A piecewise function is defined by different expressions depending on the input value.

• Example: $k(x) = \begin{cases} x+2, & x \leq -1 \\ 2x, & -1 < x \leq 1 \\ x-1, & x > 1 \end{cases}$

• To evaluate $k(x)$, determine which condition $x$ satisfies and use the corresponding formula.

Function Operations and Composition

Operations on Functions

Functions can be combined using 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: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$

Composition of Functions

The composition of two functions $f$ and $g$ is written as $(f \circ g)(x) = f(g(x))$. This means you first apply $g$ to $x$, then apply $f$ to the result.

• Domain of Composite Function: The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

• Example: If $f(x) = \sqrt{x}$ and $g(x) = x+2$, then $(f \circ g)(x) = f(g(x)) = \sqrt{x+2}$, and the domain is $x+2 \geq 0 \implies x \geq -2$.

Domain Restrictions

Common Domain Restrictions

• Even Roots: For $f(x) = \sqrt{x}$, require $x \geq 0$.

• Denominators: For $f(x) = \frac{1}{x}$, require $x \neq 0$.

• Logarithms: For $f(x) = \log(x)$, require $x > 0$.

Even and Odd Functions

Definitions and Symmetry

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric about the origin.

• Neither: If neither condition holds, the function is neither even nor odd.

Type	Algebraic Test	Graphical Symmetry
Even	$f(-x) = f(x)$	y-axis
Odd	$f(-x) = -f(x)$	Origin
Neither	Neither property holds	No symmetry

• Example (Even): $f(x) = x^2$ is even because $f(-x) = (-x)^2 = x^2 = f(x)$.

• Example (Odd): $f(x) = x^3$ is odd because $f(-x) = (-x)^3 = -x^3 = -f(x)$.

Tables and Graphs of Functions

Using Tables

Tables can be used to list input-output pairs for functions. For example:

x	f(x)	g(x)
-2	1	3
-1	0	2
0	-1	1
1	2	0
2	3	-1

To evaluate $f(g(0))$, find $g(0)$, then use that value as the input for $f$.

Using Graphs

Graphs visually represent functions and can be used to estimate values, identify domain and range, and analyze behavior such as symmetry or discontinuities.

Summary Table: Function Properties and Composition

Property	Description	Example
Domain	All valid input values	$x \geq 0$ for $f(x) = \sqrt{x}$
Range	All possible output values	$y \geq 0$ for $f(x) = x^2$
Composition	Applying one function to the result of another	$(f \circ g)(x) = f(g(x))$
Even/Odd	Symmetry properties	Even: $x^2$, Odd: $x^3$

Practice Problems (with Context)

• Given $f(x) = x^2 - 1$ and $g(x) = \sqrt{x+2}$, find $(f \circ g)(x)$ and state the domain.

• Determine if $h(x) = x^4 - x^2$ is even, odd, or neither.

• Given a piecewise function, evaluate it at several points and describe its domain and range.

Additional info: The above notes expand on the worksheet's focus on function evaluation, domain/range, composition, and symmetry, providing definitions, examples, and tables for clarity.