Precalculus Study Guide: Functions, Equations, and Graphs

Functions and Their Graphs

Function Basics

Functions are fundamental objects in precalculus, describing relationships between variables. Understanding their properties and how to represent them graphically is essential.

• Definition: A function is a rule that assigns to each element in a set called the domain exactly one element in a set called the range.

• Function Notation: If $f$ is a function, $f(x)$ denotes the output when $x$ is the input.

• Domain and Range: The domain is the set of all possible inputs; the range is the set of all possible outputs.

• Composite Functions: The composition $f(g(x))$ means applying $g$ first, then $f$.

• Inverse Functions: If $f$ is invertible, $f^{-1}(x)$ reverses the action of $f$.

• Difference Quotient: Used to measure the average rate of change: $$ \frac{f(x+h) - f(x)}{h}$$

• Properties: Functions can be increasing, decreasing, have maximum or minimum values.

Graphing Functions

• Transformations: Shifting, stretching, compressing, and reflecting graphs to obtain new functions.

• Sketching: Identify key features such as intercepts, asymptotes, and symmetry.

• Example: The graph of $f(x) = (x-2)^2 + 3$ is a parabola shifted right by 2 and up by 3.

Polynomial and Rational Functions

Polynomial Functions

Polynomial functions are sums of powers of $x$ with constant coefficients. Their graphs and zeros are central to precalculus.

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0$

• Zeros: Solutions to $f(x) = 0$ are called roots or zeros.

• Vertex: For quadratics $f(x) = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$.

• Sketching: Identify end behavior, intercepts, and turning points.

Rational Functions

• Definition: A rational function is a ratio of two polynomials: $f(x) = \frac{P(x)}{Q(x)}$.

• Vertical Asymptotes: Occur where $Q(x) = 0$ and $P(x) \neq 0$.

• Zeros: Occur where $P(x) = 0$ and $Q(x) \neq 0$.

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

Radicals and Rational Expressions

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. Simplifying and solving equations with radicals is a key skill.

• Definition: $\sqrt[n]{x}$ denotes the nth root of $x$.

• Simplifying: Combine like terms and rationalize denominators when necessary.

• Solving Radical Equations: Isolate the radical, then raise both sides to the appropriate power.

• Example: Solve $\sqrt{x+3} = 5$:

  • Square both sides: $x+3 = 25$

  • So $x = 22$

Rational Expressions

• Definition: An expression of the form $\frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

• Simplifying: Factor numerator and denominator, then reduce common factors.

• Operations: Addition, subtraction, multiplication, and division follow the rules for fractions.

Complex Numbers

Definition and Operations

Complex numbers extend the real numbers to include solutions to equations like $x^2 + 1 = 0$.

• Definition: $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, $i^2 = -1$.

• Operations: Add, subtract, multiply, and divide using algebraic rules and $i^2 = -1$.

• Example: $(2 + 3i) + (1 - 4i) = 3 - i$

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions model growth and decay processes and have the form $f(x) = a \cdot b^x$.

• Properties: $b > 0$, $b \neq 1$; the function is always positive.

• Graphing: Exponential growth if $b > 1$, decay if $0 < b < 1$.

• Example: $f(x) = 2^x$ grows rapidly as $x$ increases.

Logarithmic Functions

• Definition: $y = \log_b x$ is the inverse of $y = b^x$.

• Properties: $\log_b (xy) = \log_b x + \log_b y$; $\log_b (x^r) = r \log_b x$.

• Solving Equations: Use properties of logarithms to combine or expand expressions, then solve for the variable.

• Example: Solve $2^x = 8$:

  • Take $\log_2$ of both sides: $x = 3$

Equations and Inequalities

Quadratic, Polynomial, Rational, Radical, and Absolute Value Equations

Solving various types of equations is a core skill in precalculus.

• Quadratic Equations: $ax^2 + bx + c = 0$ can be solved by factoring, completing the square, or the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Polynomial Equations: Set the polynomial equal to zero and factor or use synthetic division.

• Rational Equations: Multiply both sides by the least common denominator to clear fractions.

• Radical Equations: Isolate the radical and raise both sides to a power to eliminate it.

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

• Inequalities: Solve similarly to equations, but consider the direction of the inequality when multiplying or dividing by negatives.

Applications and Technology

Applications

Precalculus concepts are used to solve real-world problems, such as modeling population growth, financial calculations, and physics problems.

• Example: Exponential growth models, such as $P(t) = P_0 e^{rt}$ for population growth.

• Quadratic Applications: Projectile motion, area problems, and optimization.

Technology

• Graphing Calculators: Used to plot functions, find zeros, and analyze graphs.

• Software: Tools like Desmos or GeoGebra enhance understanding of function behavior.