Functions and Their Properties: Precalculus Study Notes

Functions and Relations

Definition and Identification of Functions

A function is a relation in which each input (domain value) is paired with exactly one output (range value). Functions can be represented using tables, graphs, mappings, or equations.

• Relation: Any set of ordered pairs (x, y).

• Function: A relation where each x-value has only one corresponding y-value.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

Example: The relation {(3, 6), (6, 9), (12, 72)} is a function because each x-value is paired with only one y-value.

Non-Example: If a mapping shows one input paired with multiple outputs, it is not a function.

Determining Functions from Mappings and Sets

• Check if any input is paired with more than one output.

• If so, the relation is not a function.

• If each input has only one output, it is a function.

Example: Mapping: Alice → cat, Brad → dog, Carl → cat. This is a function if each person is paired with only one pet.

Equations and Functions

Determining if an Equation Defines y as a Function of x

To determine if an equation defines y as a function of x, solve for y and check if each x-value yields only one y-value.

• Linear equations (e.g., $y = 3x$) always define y as a function of x.

• Quadratic equations (e.g., $y^2 = x$) may not define y as a function of x because some x-values yield two y-values.

• Absolute value equations (e.g., $y = |x|$) define y as a function of x.

• Equations with even powers of y (e.g., $y^2 = x$) are not functions.

Example: $y = x^2$ is a function; $y^2 = x$ is not a function.

Evaluating Functions

Finding the Value of a Function

To evaluate a function, substitute the given value into the function's formula.

• Notation: $f(x)$ means "the value of function f at x".

• Example: If $f(x) = \frac{3x^2 - 8}{x^2 + 3}$, then $f(4) = \frac{3(4)^2 - 8}{(4)^2 + 3} = \frac{48 - 8}{16 + 3} = \frac{40}{19}$.

Function Composition and Difference Quotient

Function composition involves substituting one function into another. The difference quotient is used to measure the average rate of change of a function.

• Composition: $(f \circ g)(x) = f(g(x))$

• Difference Quotient: $\frac{f(x + h) - f(x)}{h}$, $h \neq 0$

• Example: If $f(x) = x^2$, then $\frac{f(x + h) - f(x)}{h} = \frac{(x + h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Domain and Range of Functions

Finding the Domain

The domain of a function is the set of all input values for which the function is defined.

• For rational functions, exclude values that make the denominator zero.

• For square root functions, exclude values that make the radicand negative.

• For even roots, the radicand must be non-negative.

Example: For $f(x) = \frac{3x}{x^2 - 4}$, the domain is all real numbers except $x = 2$ and $x = -2$.

Finding the Range

The range is the set of all possible output values. It can be found by analyzing the function's behavior or graph.

• For quadratic functions, the range is determined by the vertex and direction of opening.

• For rational functions, consider asymptotes and excluded values.

Function Operations

Sum, Difference, Product, and Quotient of Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• 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: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

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

Tables: Function Properties and Classification

Table: Function vs. Not a Function

Relation	Function?	Domain	Range
{(3, 6), (6, 9), (12, 72)}	Yes	3, 6, 12	6, 9, 72
Mapping: Alice → cat, Brad → dog, Carl → cat	Yes	Alice, Brad, Carl	cat, dog
Mapping: Alice → cat, Brad → cat, Carl → dog	Yes	Alice, Brad, Carl	cat, dog
{(-3, 1), (-1, 6), (2, 3)}	Yes	-3, -1, 2	1, 6, 3
{(2, 5), (1, -2), (1, 0), (1, 3)}	No	2, 1	5, -2, 0, 3
{(4, 888), (5.48, 4.888), (5.5, -6), (7, 2.33)}	Yes	4, 5.48, 5.5, 7	888, 4.888, -6, 2.33

Table: Equations and Function Status

Equation	Is y a function of x?
$y = 3x$	Yes
$9y = \frac{1}{x}$	No
$10y = |x|$	Yes
$y^2 = x + 4$	No
$y = 5x^2 - 7x + 7$	Yes

Applications: Height of a Rock Dropped from a Height

Modeling with Functions

Functions can model real-world phenomena, such as the height of a rock dropped from a certain height.

• Formula: $H(t) = h_0 - 4.9t^2$, where $H$ is height in meters, $h_0$ is initial height, and $t$ is time in seconds.

• Example: If a rock is dropped from 80 meters, after 2 seconds: $H(2) = 80 - 4.9(2)^2 = 80 - 19.6 = 60.4$ meters.

Summary

• Functions pair each input with exactly one output.

• Domains and ranges describe the set of possible inputs and outputs.

• Equations may or may not define y as a function of x.

• Function operations and composition are key concepts in precalculus.

• Applications include modeling physical phenomena with functions.