College Algebra Study Guide: Functions and Linear Models

Modeling in Mathematics

Application Problems and Mathematical Modeling

Mathematical modeling involves translating real-world situations into mathematical equations to analyze and solve problems. This process is fundamental in college algebra, especially when dealing with application problems.

• Defining Variables: Clearly identify what each variable represents in the context of the problem.

• Setting Up Equations: Use relationships described in the problem to write equations involving the defined variables.

• Solving Equations: Apply algebraic techniques to solve for the unknowns.

• Contextualizing Answers: Interpret the solution in terms of the original problem, ensuring the answer makes sense in context.

Example: If a problem states, "A rental car costs $30 per day plus $0.20 per mile driven," let x be the number of miles driven. The total cost C can be modeled as $C = 30 + 0.20x$.

Basics of All Functions

Definition and Identification of Functions

A function is a relation in which each input (domain value) corresponds to exactly one output (range value). Functions can be represented by equations, graphs, or tables.

• Graphical Identification: Use the vertical line test—if any vertical line crosses the graph more than once, it is not a function.

• Equation Identification: For an equation, check if each input yields only one output.

Example: $y = x^2$ is a function; $x^2 + y^2 = 1$ is not (it describes a circle).

Domain and Range

The domain of a function is the set of all possible input values, and the range is the set of all possible output values.

• Set-Builder Notation: $\{ x \mid x > 0 \}$

• Interval Notation: $(0, \infty)$

Example: For $f(x) = \sqrt{x}$, domain is $[0, \infty)$.

Intervals of Increasing, Decreasing, or Constant

Functions can be classified based on their behavior over intervals:

• Increasing: Function values rise as input increases.

• Decreasing: Function values fall as input increases.

• Constant: Function values remain unchanged.

Use interval notation to describe these intervals, e.g., "f(x) is increasing on $(1, 3)$".

Relative Minimums and Maximums

A relative minimum is a point where the function value is lower than at nearby points; a relative maximum is higher than at nearby points.

• Identify these points graphically or algebraically.

• State both the location (input value) and the value (output).

Example: For $f(x) = x^2$, the minimum is at $x = 0$, $f(0) = 0$.

Evaluating Functions

Functions can be evaluated by substituting input values into the function rule.

• Graphical Evaluation: Find the output value for a given input on the graph.

• Algebraic Evaluation: Substitute the input into the function's formula.

Example: If $f(x) = 2x + 1$, then $f(3) = 2 \times 3 + 1 = 7$.

Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry.

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

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

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Linear Functions

Finding Linear Functions from Points

A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Given two points $(x_1, y_1)$ and $(x_2, y_2)$, calculate the slope:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Use the slope and one point to find $b$:

$b = y_1 - m x_1$

Example: Points (1, 2) and (3, 6): $m = \frac{6-2}{3-1} = 2$, $b = 2 - 2 \times 1 = 0$, so $f(x) = 2x$.

Linear Functions from Parallel or Perpendicular Lines

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Parallel: If a line has slope $m$, a parallel line also has slope $m$.

• Perpendicular: If a line has slope $m$, a perpendicular line has slope $-\frac{1}{m}$.

• Use the given point to find the y-intercept.

Example: If a line has slope 3, a perpendicular line has slope $-\frac{1}{3}$.

Slope, Y-Intercept, and X-Intercept

For a linear equation $y = mx + b$:

• Slope ($m$): Rate of change of $y$ with respect to $x$.

• Y-Intercept ($b$): Value of $y$ when $x = 0$.

• X-Intercept: Value of $x$ when $y = 0$; solve $0 = mx + b$ for $x$.

Example: For $y = 2x + 3$, slope is 2, y-intercept is 3, x-intercept is $x = -\frac{3}{2}$.

Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the domain.

• Graph each piece over its specified interval.

• Pay attention to open and closed endpoints.

Example: $f(x) = \begin{cases} x+1 & x < 0 \\ 2x & x \geq 0 \end{cases}$

Difference Quotient

The difference quotient is used to measure the average rate of change of a function over an interval and is foundational in calculus.

• For a function $f(x)$, the difference quotient is:

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

Example: For $f(x) = x^2$, $\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$.