Functions and Linear Functions: Definitions, Properties, and Applications

Functions & Graphs

Definition of a Function

A function is a special type of relation in mathematics where each input from the domain is associated with exactly one output in the range. Functions are fundamental in precalculus, as they model relationships between variables.

• Domain: The set of all possible input values (usually denoted as x).

• Range: The set of all possible output values (usually denoted as f(x)).

• Well-defined function: Each input has a single, unique output.

• Not a function: If an input is associated with more than one output, the relation is not a function.

Example: If $f(1) = 2$ and $f(2) = 3$, this is a function. If $f(2)$ is assigned both 3 and 4, it is not well-defined and not a function.

Vertical Line Test

The vertical line test is a graphical method to determine if a curve represents a function. If any vertical line crosses the graph more than once, the relation is not a function.

• Application: Useful for quickly checking if a graph is a function.

Notation and Examples

• Function notation: $f(x)$ denotes the output for input $x$.

• Example: $f(x) = 2x + 1$

• Evaluating: $f(4) = 2 \times 4 + 1 = 9$

Special Functions

• Prime Counting Function: $\pi(x)$ counts the number of prime numbers less than or equal to $x$.

• Example values:

  • $\pi(2) = 1$

  • $\pi(3) = 2$

  • $\pi(4) = 2$

  • $\pi(5) = 3$

• Approximation: $\pi(x) \approx \frac{x}{\ln x}$

Linear Functions

Definition and Properties

A linear function is a function whose graph is a straight line. It can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope ($m$): Measures the steepness of the line.

• Y-intercept ($b$): The value of $y$ when $x = 0$.

• General form: $y = mx + b$

Finding the Equation of a Line

• Given a point and slope: Use the point-slope form $y - y_1 = m(x - x_1)$

• Given two points: Find the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, then use point-slope form.

Example: Find the line through the x-intercept of $y = 9x - 18$ parallel to $y = \frac{46x}{5}$.

• Step 1: Find the x-intercept of $y = 9x - 18$ by setting $y = 0$:

  • $0 = 9x - 18 \Rightarrow x = 2$

• Step 2: The slope of $y = \frac{46x}{5}$ is $\frac{46}{5}$.

• Step 3: Use point-slope form with point $(2, 0)$:

  • $y = \frac{46}{5}(x - 2)$

Parallel and Perpendicular Lines

• Parallel lines: Have the same slope.

• Perpendicular lines: Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$

Table: Linear Function Properties

Form	Slope ($m$)	Y-intercept ($b$)	Example
$y = mx + b$	$m$	$b$	$y = 2x + 3$
Point-slope: $y - y_1 = m(x - x_1)$	$m$	Through $(x_1, y_1)$	$y - 4 = 5(x - 2)$

Additional info:

• Some content was inferred from context, such as the use of the prime counting function and the vertical line test.

• Examples and formulas were expanded for clarity and completeness.