Comprehensive Study Guide for College Algebra Final Exam

Piecewise Functions and Graphs

Understanding Piecewise-Defined Functions

Piecewise functions are defined by different expressions over different intervals of the domain. Their graphs may include open or closed circles at endpoints and may assign special values at specific points.

• Formula on Each Interval: Each piece of the function applies to a specific interval.

• Endpoints: Open circles indicate the endpoint is not included; closed circles indicate inclusion.

• Special Values: Sometimes a function assigns a value at a single point, distinct from the intervals.

• Types of Pieces: Pieces may be linear, constant, or nonlinear (e.g., quadratic).

• Example: For $h(x) = \begin{cases} 2x-1, & -4 \leq x < 1 \\ -5, & x=1 \\ -x+3, & x>1 \end{cases}$, identify the graph by checking endpoints and the special value at $x=1$.

Exponent Rules and Negative Exponents

Simplifying Expressions with Exponents

Exponent rules allow simplification of expressions involving powers, products, quotients, and negative exponents. Final answers should use only positive exponents.

• Product Rule: $(ab)^n = a^n b^n$

• Power of a Power: $(x^m)^n = x^{mn}$

• Quotient Rule: $\frac{x^m}{x^n} = x^{m-n}$

• Negative Exponent: $x^{-n} = \frac{1}{x^n}$

• Example: Simplify $\frac{(-3x^{-2}y^{-1})^2}{(6x^{-3}y)^3}$ and write with positive exponents only.

Rational Expressions and Excluded Values

Combining and Simplifying Rational Expressions

When adding or subtracting rational expressions, combine numerators and keep track of values that make any denominator zero (excluded values).

• Excluded Values: Any value that makes a denominator zero must be excluded from the domain, even if it cancels later.

• Example: Simplify $\frac{4x-5}{x^2-16} - \frac{x-9}{x^2-16}$ and state all excluded values.

Complex Numbers

Simplifying Expressions with Imaginary Numbers

Complex numbers involve the imaginary unit $i$, where $i^2 = -1$. Square roots of negative numbers are rewritten using $i$.

• Definition: $\sqrt{-a} = i\sqrt{a}$ for $a > 0$.

• Standard Form: Write complex numbers as $a + bi$.

• Example: Simplify $(4 - \sqrt{-49})^2$ and write in standard form.

Rational Equations, Restrictions, and Extraneous Solutions

Solving Rational Equations

Before solving, state all values that make denominators zero. After solving, check that solutions are not excluded or extraneous.

• Restrictions: List all values that make any denominator zero.

• Solving: Multiply both sides by the least common denominator (LCD), solve, and check solutions.

• Example: Solve $\frac{2}{x+4} + \frac{3}{x-4} = \frac{22}{(x+4)(x-4)}$ and state restrictions.

Quadratic Equations and the Discriminant

Solving Quadratics and Analyzing Solutions

The discriminant $D = b^2 - 4ac$ determines the number and type of solutions for $ax^2 + bx + c = 0$.

• Discriminant: $D > 0$ (two real solutions), $D = 0$ (one real solution), $D < 0$ (two complex solutions).

• Example: Solve $2x^2 - 7x = 15$ and state the discriminant.

Domains of Quotients of Functions

Finding the Domain of a Quotient

The domain of $\frac{f}{g}(x)$ is all real numbers except where $g(x) = 0$.

• Procedure: Solve $g(x) = 0$ and exclude those values from the domain.

• Example: For $f(x) = 5x^2 - 10x$, $g(x) = x^2 + 4x - 12$, find the domain of $\frac{f}{g}(x)$ in interval notation.

Number of Solutions of a Linear System

Classifying Solutions to Linear Systems

A system of two linear equations can have one solution, no solution, or infinitely many solutions, depending on the slopes and intercepts.

• One Solution: Different slopes.

• No Solution: Same slope, different intercepts (parallel lines).

• Infinitely Many Solutions: Same line (identical equations).

• Example: Determine the number of solutions for $\begin{cases} y = -2x + 5 \\ 4x + 2y = 8 \end{cases}$.

Domains of Natural Logarithmic Functions

Finding the Domain of Logarithmic Functions

The argument of a natural logarithm must be positive: $A > 0$ for $\ln(A)$.

• Procedure: Set the argument greater than zero and solve for $x$.

• Example: Find the domain of $f(x) = \ln\left(\frac{1}{x} - 5\right)$.

Exponential Equations with Different Bases

Solving Exponential Equations Using Logarithms

When bases are different, take logarithms of both sides to solve for the variable.

• Logarithm Rule: $\ln(a^u) = u\ln(a)$

• Example: Solve $3^{x-2} = 7^{2x+1}$ for $x$ in logarithmic form and as a decimal.

Reading a Graph of a Function

Interpreting Function Graphs

Graphs provide information about domain, range, intercepts, intervals of increase/decrease, relative extrema, and function values.

• Domain and Range: Set of possible $x$ and $y$ values, respectively.

• Intercepts: $x$-intercepts (where $y=0$), $y$-intercept (where $x=0$).

• Increasing/Decreasing: Intervals where the function rises or falls.

• Relative Extrema: Local maximum or minimum points.

• Example: For $g(x) = x^3 - 3x + 1$, use the graph to answer questions about domain, range, intercepts, intervals, and function values.

Composition Using a Graph or Formula

Evaluating Function Compositions

For $(f \circ g)(x)$, first compute $g(x)$, then use that result as the input for $f$.

• Procedure: Compute $g(x)$, then $f(g(x))$.

• Example: If $f(x) = x^2 - 4x + 1$ and $g(x) = x + 3$, find $(f \circ g)(-1)$.

One-to-One Functions and Inverses

Determining Invertibility of Functions

A function has an inverse if it is one-to-one (passes the horizontal line test). Sometimes, restricting the domain allows for an inverse.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Domain Restriction: Limiting the domain can make a function invertible.

• Example: For $k(x) = (x-2)^2$, explain why it does not have an inverse on $\mathbb{R}$, and give a restricted domain where it does.

Restricted Domains and Inverse Values

Finding Inverse Values with Restricted Domains

When a function's domain is restricted, inverse values must come from that interval. To find $f^{-1}(a)$, solve $f(x) = a$ for $x$ in the restricted domain.

• Example: If $f(x) = x^2 - 4x + 1$ with domain $[2, \infty)$, estimate or compute $f^{-1}(10)$.

Rational Function Composition and Inverses

Composing and Inverting Rational Functions

To find $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$ and simplify. To find the inverse, swap $x$ and $y$ and solve for $y$.

• Domain Restrictions: State values excluded from the domain after composition.

• Example: For $f(x) = \frac{1}{x-4}$ and $g(x) = \frac{12}{x}$, find $(f \circ g)(x)$, its domain, and $f^{-1}(x)$.

Expanding Logarithmic Expressions

Using Logarithm Properties to Expand Expressions

Apply product, quotient, and power rules to expand logarithmic expressions. Rewrite radicals as fractional exponents.

• Product Rule: $\log(AB) = \log A + \log B$

• Quotient Rule: $\log\left(\frac{A}{B}\right) = \log A - \log B$

• Power Rule: $\log(A^r) = r\log A$

• Example: Expand $\log\left(\frac{5x^3\sqrt{x}-1}{2(x+4)^3}\right)$ fully.

Exponential Decay Models

Evaluating Exponential Decay

Exponential decay is modeled by $A = A_0 e^{kt}$ with $k < 0$. Substitute the given value of $t$ to find the remaining amount.

• Example: For $A = 600e^{-0.004t}$, find $A$ when $t = 120$ years.

Solving Simple Exponential Equations

Solving for the Exponent

For equations like $a^x = b$, take logarithms to solve for $x$.

• Formula: $x = \frac{\ln b}{\ln a}$

• Example: Solve $0.8^x = 12$ and approximate to four decimal places.

Systems of Three Linear Equations

Solving Three-Variable Systems

Use elimination or substitution to reduce the system to two variables, then one. Back-substitute to find all variables.

• Example: Solve $\begin{cases} x + y + z = 6 \\ 2x - y + 3z = 11 \\ -x + 4y + z = 4 \end{cases}$.

Logarithmic Equations

Solving Equations Involving Logarithms

Impose domain restrictions, use log rules, and check solutions in the original equation.

• Example: Solve $\ln(x-3) - \ln(x+3) = \ln(x-1) - \ln(x+5)$, state domain restrictions, and check solutions.

Graphs of $e^x$ and $\ln x$

Understanding the Relationship Between Exponential and Logarithmic Functions

The graphs of $y = e^x$ and $y = \ln x$ are reflections across $y = x$. Their domains and ranges are complementary.

• $e^x$: Domain $(-\infty, \infty)$, Range $(0, \infty)$

• $\ln x$: Domain $(0, \infty)$, Range $(-\infty, \infty)$

• Vertical Asymptote: $\ln x$ has a vertical asymptote at $x=0$.

• Example: Sketch $y = e^x$, $y = \ln x$, and $y = x$ on the same axes. Explain their inverse relationship.

Slope and y-Intercept

Identifying Slope and Intercept from Linear Equations

Rewrite the equation in $y = mx + b$ form to identify the slope ($m$) and $y$-intercept ($b$).

• Example: For $3x - 4y + 12 = 0$, solve for $y$ and identify $m$ and $b$.

Polynomial Expressions from Area Models

Finding the Area of a Shaded Region

Subtract the area of the unshaded region from the larger figure and simplify the resulting polynomial.

• Example: A large rectangle has dimensions $x+7$ by $x+5$, and a smaller rectangle inside has dimensions $x+3$ by $x+1$. Write a polynomial for the shaded area.

Quadratic Models from Data Points

Setting Up a System for a Quadratic Model

Each data point $(x, y)$ substituted into $y = ax^2 + bx + c$ gives one equation. Three points yield a system of three equations.

• Example: For points $(1, 20)$, $(2, 27)$, $(4, 15)$, write the system for $y = ax^2 + bx + c$.

Factoring Strategy

Choosing the First Factoring Method

Common factoring methods include GCF, difference of squares, perfect square trinomials, grouping, trinomials by trial and error, and difference of cubes. Always check for a GCF first unless otherwise specified.

• Factoring Methods:

  • GCF: Greatest Common Factor

  • DOS: Difference of Squares

  • PST: Perfect Square Trinomial

  • GRP: Grouping

  • TRI: Trinomial by Trial and Error

  • DOC: Difference of Cubes

• Example: For each expression, identify the first nontrivial factoring method:

  1. $18x^4 - 12x^2$

  2. $16x^2 - 81$

  3. $x^2 + 14x + 49$

  4. $3x^3 + 6x^2 + 5x + 10$

  5. $2x^2 + 7x - 15$

  6. $64x^3 - 27$

Final Skills Checklist

Essential Skills for the Final Exam

• Graph and identify piecewise-defined functions, including endpoint behavior.

• Simplify expressions using exponent rules and write with positive exponents.

• Combine rational expressions and state all excluded values.

• Simplify complex-number expressions and write in $a + bi$ form.

• Solve rational equations, list restrictions, and reject extraneous solutions.

• Solve quadratic equations and compute the discriminant $D = b^2 - 4ac$.

• Find domains of quotient and logarithmic functions.

• Classify solutions to linear systems (one, none, or infinitely many).

• Solve exponential equations using logarithms and evaluate exponential decay models.

• Interpret graphs for domain, range, intercepts, intervals, extrema, and function values.

• Compute compositions $(f \circ g)(x)$ using formulas or graphs.

• Determine if a function has an inverse using the horizontal line test.

• Work with inverse values for restricted domains.

• Find the inverse of a rational function by interchanging $x$ and $y$ and solving for $y$.

• Expand logarithmic expressions using log rules.

• Solve logarithmic equations and check domain restrictions.

• Recognize the relationship between $y = e^x$ and $y = \ln x$.

• Rewrite linear equations in slope-intercept form and identify slope and intercept.

• Write polynomials for shaded areas by subtracting and simplifying.

• Set up systems for quadratic models from data points.

• Choose appropriate factoring strategies for given expressions.