MAC1105 College Algebra: Structured Study Guide Based on Course Schedule

Functions

Introduction to Functions, Domain, and Range

Functions are fundamental objects in algebra that describe relationships between sets. Understanding their domain and range is essential for analyzing their behavior.

• Function: A rule that assigns each element in a set (domain) to exactly one element in another set (range).

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

• Range: The set of all possible output values (y-values) produced by the function.

• Example: For $f(x) = x^2$, the domain is all real numbers, and the range is all non-negative real numbers.

Function Behaviors and Properties

Functions can be classified and analyzed based on their behaviors, such as increasing, decreasing, and constant intervals.

• Increasing Function: A function is increasing on an interval if, as x increases, f(x) increases.

• Decreasing Function: A function is decreasing on an interval if, as x increases, f(x) decreases.

• Constant Function: A function is constant on an interval if f(x) remains the same as x changes.

• Example: $f(x) = 2x + 3$ is increasing everywhere.

Linear vs. Non-Linear Functions and Transformations

Linear functions have constant rates of change, while non-linear functions do not. Transformations modify functions by shifting, stretching, or reflecting their graphs.

• Linear Function: Has the form $f(x) = mx + b$.

• Non-Linear Function: Examples include quadratic, cubic, exponential, etc.

• Transformations: Include translations, reflections, stretches, and compressions.

• Example: $f(x) = x^2$ shifted up by 3 units: $f(x) = x^2 + 3$.

Variations

Variation describes how one quantity changes in relation to another. Common types include direct, inverse, and joint variation.

• Direct Variation: $y = kx$

• Inverse Variation: $y = \frac{k}{x}$

• Joint Variation: $y = kxz$

• Example: If $y$ varies directly with $x$, and $k=2$, then $y=2x$.

Linear and Quadratic Functions

Linear Functions

Linear functions model constant rates of change and are represented by straight lines.

• General Form: $f(x) = mx + b$

• Slope (m): Measures the rate of change.

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

• Example: $f(x) = 4x - 1$

Quadratic Functions and Their Zeros

Quadratic functions are polynomials of degree 2 and their zeros are the solutions to $ax^2 + bx + c = 0$.

• Standard Form: $f(x) = ax^2 + bx + c$

• Zeros: Values of $x$ where $f(x) = 0$

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: $f(x) = x^2 - 5x + 6$ has zeros at $x=2$ and $x=3$.

Quadratic Function Properties and Modeling

Quadratic functions have parabolic graphs and are used in modeling various real-world phenomena.

• Vertex: The highest or lowest point of the parabola, given by $x = \frac{-b}{2a}$

• Axis of Symmetry: Vertical line $x = \frac{-b}{2a}$

• Example: Projectile motion modeled by $h(t) = -16t^2 + v_0t + h_0$

Complex Zeros and Absolute Value Equations/Inequalities

Quadratic equations may have complex zeros if the discriminant is negative. Absolute value equations and inequalities involve expressions within absolute value bars.

• Complex Zeros: Occur when $b^2 - 4ac < 0$

• Absolute Value Equation: $|x| = a$ has solutions $x = a$ or $x = -a$

• Absolute Value Inequality: $|x| < a$ implies $-a < x < a$

• Example: $x^2 + 4 = 0$ has zeros $x = 2i$ and $x = -2i$

Systems of Equations

Systems of Linear and Non-Linear Equations

Systems of equations involve solving for variables that satisfy multiple equations simultaneously.

• Linear System: All equations are linear.

• Non-Linear System: At least one equation is non-linear.

• Methods: Substitution, elimination, and graphical methods.

• Example: Solve $x + y = 5$ and $x - y = 1$.

Polynomial and Rational Functions

Polynomial and Power Functions

Polynomial functions are sums of powers of x with coefficients. Power functions have the form $f(x) = ax^n$.

• Polynomial Function: $f(x) = a_nx^n + ... + a_1x + a_0$

• Power Function: $f(x) = ax^n$

• Example: $f(x) = 3x^4 - 2x^2 + 7$

Rational Functions and Their Attributes

Rational functions are ratios of polynomials. Their attributes include domain, asymptotes, and intercepts.

• Rational Function: $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$

• Vertical Asymptote: Occurs where $Q(x) = 0$

• Horizontal Asymptote: Determined by degrees of $P(x)$ and $Q(x)$

• Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$

Rational Functions Analysis

Analyzing rational functions involves finding intercepts, asymptotes, and behavior near undefined points.

• Find Domain: Exclude values where denominator is zero.

• Find Asymptotes: Set denominator to zero for vertical, compare degrees for horizontal.

• Example: $f(x) = \frac{x+1}{x-3}$

Composite, Inverse, Exponential, and Logarithmic Functions

Composite Functions

Composite functions combine two functions, applying one after the other.

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

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

Invertibility and Inverse Functions

A function is invertible if each output corresponds to exactly one input. The inverse function reverses the original mapping.

• Inverse Function: $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$

• Test for Invertibility: Function must be one-to-one (pass the horizontal line test).

• Example: $f(x) = 2x + 3$ has inverse $f^{-1}(x) = \frac{x-3}{2}$

Exponential Functions

Exponential functions have the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Growth: If $a > 1$, function increases rapidly.

• Decay: If $0 < a < 1$, function decreases rapidly.

• Example: $f(x) = 2^x$

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions.

• General Form: $f(x) = \log_a(x)$

• Properties: $\log_a(a^x) = x$ and $a^{\log_a(x)} = x$

• Example: $f(x) = \log_{10}(x)$

Applications of Exponential and Logarithmic Functions

These functions are used in modeling growth, decay, and other real-world phenomena.

• Exponential Growth: $N(t) = N_0 e^{rt}$

• Exponential Decay: $N(t) = N_0 e^{-rt}$

• Logarithmic Applications: pH in chemistry, Richter scale for earthquakes

• Example: Population growth modeled by $P(t) = P_0 e^{kt}$

Review and Exam Preparation

Review Sessions

Review sessions are scheduled before exams to reinforce concepts and practice problem-solving.

• Key Strategies: Practice with sample problems, review formulas, and clarify concepts.

• Example: Review quadratic formula and its applications.

Exam Days

Exams assess understanding of course material, including functions, equations, and applications.

• Preparation: Review all major topics, practice solving equations, and understand function properties.

• Example: Exam may include solving systems of equations and analyzing function graphs.