Precalculus Study Notes: Functions and Their Graphs (Section 1.1)

Functions and Their Graphs

Section 1.1: Functions

This section introduces the foundational concept of functions in mathematics, focusing on their definitions, properties, and how to work with them. Understanding functions is essential for further study in precalculus and calculus.

Objectives

• Describe a relation

• Determine whether a relation represents a function

• Use function notation and evaluate functions

• Compute the difference quotient of a function

• Find the domain of a function defined by an equation

• Perform operations (sum, difference, product, quotient) on functions

1. Relations

A relation is a correspondence between two sets, where each element of the first set is paired with one or more elements of the second set. In mathematics, relations are often described using ordered pairs (x, y), where x is from the first set (domain) and y is from the second set (range).

• Domain: The set of all possible first elements (inputs) in the relation.

• Range: The set of all possible second elements (outputs) in the relation.

Example: The relation between states and the number of representatives in the House of Representatives can be written as a set of ordered pairs:

• (Mississippi, 4), (Louisiana, 6), (Alabama, 7), (Georgia, 14), (Florida, 27)

Domain: {Mississippi, Louisiana, Alabama, Georgia, Florida} Range: {4, 6, 7, 14, 27}

2. Functions

A function is a special type of relation in which each element of the domain is paired with exactly one element of the range. In other words, for every input, there is only one output.

• Notation: If f is a function from X to Y, we write f: X → Y.

• Each input x in the domain corresponds to a unique output y in the range, written as y = f(x).

Example: If X = {Maziply, Amazon, Primegood2, Toys & Co., kids create fun, GoodLocker} and Y = {$9.99, $10.80, $10.99, $11.83, $14.99}, a function assigns each store exactly one price.

Determining if a Relation is a Function

• If any input (first element) is paired with more than one output (second element), the relation is not a function.

• If every input is paired with exactly one output, the relation is a function.

Example: The set {(1,5), (3,9), (5,1), (9,2)} is a function because no input repeats with a different output. The set {(1,1), (2,2), (3,3), (2,-2), (-5,-3)} is not a function because the input 2 is paired with both 2 and -2.

3. Function Notation and Evaluation

Functions are often written using function notation: y = f(x). Here, x is the independent variable (input), and y is the dependent variable (output).

• To evaluate a function, substitute the input value into the formula.

Example: If f(x) = 2x^2 - 5x, then

• f(3) = 2(3)^2 - 5(3) = 18 - 15 = 3

4. Explicit and Implicit Forms of Functions

• Explicit form: The function is solved for y in terms of x, e.g., y = f(x) = -2x + 4.

• Implicit form: The function is given by an equation involving both x and y, e.g., x + y = 4.

5. The Difference Quotient

The difference quotient is a formula used to compute the average rate of change of a function over an interval. It is fundamental in calculus for defining the derivative.

• The difference quotient of a function f at x is:

$\frac{f(x+h) - f(x)}{h}$

Example: For $f(x) = \sqrt{x+1}$, the difference quotient is:

$\frac{\sqrt{x+h+1} - \sqrt{x+1}}{h}$

6. Domain of a Function

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

• If the function has a denominator, exclude values that make the denominator zero.

• If the function has an even-index radical (e.g., square root), exclude values that make the radicand negative.

Example: For $F(x) = \sqrt{2x+6} / (x-2)$:

• Require $2x+6 \geq 0 \implies x \geq -3$

• Exclude $x = 2$ (denominator zero)

• Domain: $\{x \mid x \geq -3, x \neq 2\}$

Application Example: The volume of a cube as a function of its side length s is $V(s) = s^3$. The domain is $\{s \mid s > 0\}$.

7. Operations on Functions

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the resulting function is the intersection of the domains of the original functions (excluding values that make any denominator zero).

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x)g(x)$

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

Example: If $f(x) = \frac{x-4}{x}$ and $g(x) = \frac{2x-4}{x+3}$, then

• $(f+g)(x) = \frac{x-4}{x} + \frac{2x-4}{x+3}$

• The domain is all real x except $x = 0$ and $x = -3$

Summary Table: Types of Relations and Functions

Type	Description	Example
Relation	Any set of ordered pairs	{(1,2), (2,3), (2,4)}
Function	Each input has exactly one output	{(1,2), (2,3), (3,4)}
Not a Function	At least one input has more than one output	{(1,2), (1,3)}

Key Takeaways:

• A function is a relation with exactly one output for each input.

• Function notation and evaluation are essential skills.

• The domain of a function is determined by the formula and any restrictions (such as division by zero or negative radicands).

• Functions can be combined using arithmetic operations, with domains adjusted accordingly.