Functions and Linear Functions: Precalculus Study Notes

Functions

Definition and Key Concepts

A function is a mathematical relation that assigns each element from a set called the domain to exactly one element in another set called the codomain. The set of actual outputs produced by the function is called the range or image.

• Domain: The set of all possible input values (often denoted as $A$).

• Codomain: The set of all possible output values (often denoted as $B$).

• Range: The set of actual outputs (subset of the codomain).

• Notation: If $f$ is a function from $A$ to $B$, we write $f: A \rightarrow B$.

• Independent variable: The input to the function (often $x$).

• Dependent variable: The output of the function (often $y$ or $f(x)$).

Most commonly, functions are defined on the set of real numbers, denoted $\mathbb{R}$.

Why Functions Matter

• Functions model real-world phenomena, allowing predictions based on current conditions.

• Examples include population growth, physical laws, and financial models.

Ways to Represent Functions

• Pictorially: Using diagrams or mapping arrows between sets.

• Tables: Listing input-output pairs.

• Graphically: Plotting points $(x, f(x))$ in the coordinate plane.

• Formulas: Expressing the function as an equation, e.g., $f(x) = 2x + 1$.

• Verbally: Describing the function in words, e.g., "the number of primes less than or equal to $x$".

Examples of Functions

• Formula: $f(x) = 10x + 9$

• Table:

  x	f(x)
  0	1
  1	4
  6	8

• Graph: Points plotted in the coordinate plane, e.g., $(x, f(x))$.

• Wordy: $\pi(x)$ = number of primes $\leq x$; $\pi(17) = 7$.

Vertical Line Test

The vertical line test is used to determine if a graph represents a function. If every vertical line intersects the graph at most once, the relation is a function.

Operations with Functions

New functions can be built from existing ones using various operations:

• Addition: $(f + g)(x) = f(x) + g(x)$

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

• Multiplication: $(fg)(x) = f(x) \cdot g(x)$

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

• Scalar multiplication: $(cf)(x) = c \cdot f(x)$ for $c \in \mathbb{R}$

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

Examples of Function Operations

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

  • $(f \circ g)(4) = f(g(4)) = f(16) = 2 \cdot 16 + 1 = 33$

  • $(fg)(0) = f(0) \cdot g(0) = 1 \cdot 0 = 0$

  • $(2g)(1) = 2 \cdot g(1) = 2 \cdot 1 = 2$

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.

• The function has a constant average rate of change (AROC).

• The slope $m$ is calculated as: $\frac{y_2 - y_1}{x_2 - x_1} = m$

Forms of Linear Equations

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_0 = m(x - x_0)$

• General Form: $Ax + By = C$

Graphing Linear Functions

• The graph is a straight line.

• Horizontal lines: $y = c$ (slope $m = 0$)

• Vertical lines: $x = c$ (undefined slope)

• To graph a line, you need a point and the slope.

Intercepts

• x-intercept: The point where the graph crosses the x-axis ($y = 0$).

• y-intercept: The point where the graph crosses the y-axis ($x = 0$).

Parallel and Perpendicular Lines

• Parallel lines: Have equal slopes.

• Perpendicular lines: Have slopes that are negative reciprocals.

Solving Linear Equations

• Move variables to one side and constants to the other.

• Solve for the unknown variable.

• Label variables and functions appropriately in real-life contexts.

Example: Modeling Population Growth

Suppose a city's population increased from 180,000 in 2000 to 240,000 in 2020. We can model this with a linear function:

• Let $P(t)$ be the population $t$ years after 2000.

• Initial value: $P(0) = 180,000$

• After 20 years: $P(20) = 240,000$

• Average rate of change (slope): $\frac{240,000 - 180,000}{20} = 3,000$

• Linear model: $P(t) = 3,000t + 180,000$

• Estimate population in 2030 ($t = 30$): $P(30) = 3,000 \cdot 30 + 180,000 = 270,000$

• Find when population reaches 300,000: $300,000 = 3,000t + 180,000 \Rightarrow t = 40$ So, in 2040.

Additional info: The notes briefly mention the importance of labeling variables and interpreting intercepts in context. For example, the y-intercept often represents an initial value, and the x-intercept may represent an end state.