College Algebra Final Exam Study Guide – Core Topics and Examples

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: Use open circles for excluded endpoints and closed circles for included endpoints.

• Special Values: Sometimes a function assigns a value at a single point, e.g., h(x) at x = 1.

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 you to simplify expressions involving products, quotients, powers, and negative exponents.

• Product Rule: $(ab)^n = a^n b^n$

• Power Rule: $(x^m)^n = x^{mn}$

• 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

Adding and Subtracting Rational Expressions

When combining rational expressions, always consider values that make any denominator zero. These are excluded from the domain, even if they cancel later.

• Combine Numerators: Use a common denominator.

• State Excluded Values: List all values that make any denominator zero.

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 $i^2 = -1$. The square root of a negative number is written as $\sqrt{-a} = i\sqrt{a}$.

• Standard Form: $a + bi$

• Operations: Expand and simplify using $i^2 = -1$.

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

Rational Equations, Restrictions, and Extraneous Solutions

Solving Rational Equations

Before solving, identify values that make denominators zero. After solving, check that solutions are not excluded.

• Restrictions: State values that make any denominator zero.

• Least Common Denominator (LCD): Multiply both sides by the LCD to clear denominators.

• Check Solutions: Discard extraneous solutions that are not in the domain.

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 Using the Discriminant

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

• $D > 0$: Two distinct real solutions.

• $D = 0$: One real solution (double root).

• $D < 0$: 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 $g(x) = 0$.

• Set Denominator to Zero: Solve $g(x) = 0$.

• Exclude Solutions: Remove these 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.

• Different Slopes: One solution (lines intersect).

• Same Slope, Different Intercepts: No solution (parallel lines).

• Same Line: Infinitely many solutions.

Example: For $\begin{cases} y = -2x + 5 \\ 4x + 2y = 8 \end{cases}$, determine the number of solutions.

Domains of Natural Logarithmic Functions

Finding the Domain of Logarithmic Functions

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

• Solve Inequality: Set the argument $> 0$ 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

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

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

• Exact and Approximate Answers: Give answers in logarithmic form and as decimals if required.

Example: Solve $3^{x-2} = 7^{2x+1}$ for $x$.

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.

• Intercepts: Where the graph crosses axes.

• Increasing/Decreasing: 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 $(f \circ g)(x) = f(g(x))$, compute $g(x)$ first, then use that value as the input for $f$.

• Order Matters: Always evaluate the inside function first.

• Graphical Evaluation: Use the graph to find $f(g(x))$ if $f$ is given graphically.

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

A function is one-to-one if it passes the horizontal line test. Only one-to-one functions have inverses on their domain.

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

• Domain Restriction: Sometimes restricting the domain makes a function invertible.

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

Restricted Domains and Inverse Values

Finding Inverse Values with Domain Restrictions

When a function's domain is restricted, only values from that interval are used to find inverse values.

• Estimate $f^{-1}(a)$: Find $x$ in the restricted domain such that $f(x) = a$.

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

Rational Function Composition and Inverses

Composing and Inverting Rational Functions

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

• Domain Restrictions: State where the composed function is defined.

• Inverse Process: Replace $f(x)$ with $y$, swap $x$ and $y$, solve for $y$.

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

Expand logarithmic expressions using the product, quotient, and power rules. 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.

• Initial Value: $A_0$ is the starting amount.

• Decay Constant: $k$ is negative for decay.

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

Solving Simple Exponential Equations

Solving $a^x = b$

Take logarithms of both sides to solve for $x$.

• Formula: $x = \frac{\ln b}{\ln a}$

• Negative Solutions: If $0 < a < 1$ and $b > 1$, $x$ is negative.

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.

• Stepwise Reduction: Eliminate one variable at a time.

• Check Solution: Substitute back to verify the ordered triple.

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

Logarithmic Equations

Solving Equations with Logarithms

Impose domain restrictions before solving. Use log rules and check all solutions in the original equation.

• Domain Restrictions: Arguments of all logarithms must be positive.

• Log Rules: Combine or expand as needed.

• Check Solutions: Substitute back to ensure validity.

Example: Solve $\ln(x-3) - \ln(x+3) = \ln(x-1) - \ln(x+5)$ and state restrictions.

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

Relationship Between Exponential and Logarithmic Graphs

The graphs of $y = e^x$ and $y = \ln x$ are reflections across $y = x$ because they are inverse functions.

• Domain and Range: $e^x$: domain $(-\infty, \infty)$, range $(0, \infty)$; $\ln x$: domain $(0, \infty)$, range $(-\infty, \infty)$.

• Asymptotes: $\ln x$ 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

Rewrite a linear equation in the form $y = mx + b$ to identify the slope ($m$) and y-intercept ($(0, b)$).

• Slope: $m$ is the coefficient of $x$.

• y-Intercept: The value of $y$ when $x = 0$.

Example: For $3x - 4y + 12 = 0$, find the slope and y-intercept.

Polynomial Expressions from Area Models

Finding the Area of a Shaded Region

Subtract the area of the unshaded part from the area of the larger figure. Expand and combine like terms to write the result as a polynomial.

• Area of Rectangle: $A = l \times w$

• Shaded Area: $A_{shaded} = A_{large} - A_{small}$

Example: For rectangles with dimensions $(x+7)$ by $(x+5)$ and $(x+3)$ 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.

• Substitute Each Point: Write one equation per point.

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

Factoring Strategy

Choosing a Factoring Method

Common factoring methods include GCF (greatest common factor), difference of squares, perfect square trinomial, grouping, trinomial by trial and error, and difference of cubes. Always check for a GCF first unless otherwise specified.

• GCF: Factor out the greatest common factor.

• Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$

• Perfect Square Trinomial: $a^2 + 2ab + b^2 = (a+b)^2$

• Grouping: Factor by grouping terms.

• Trinomial by Trial and Error: Factor $ax^2 + bx + c$ by finding two numbers that multiply to $ac$ and add to $b$.

• Difference of Cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

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

• 18x4 −12x2 (GCF)

• 16x2 −81 (Difference of Squares)

• x2 + 14x + 49 (Perfect Square Trinomial)

• 3x3 + 6x2 + 5x + 10 (Grouping)

• 2x2 + 7x −15 (Trinomial by Trial and Error)

• 64x3 −27 (Difference of Cubes)

Final Skills Checklist

Essential Skills for College Algebra

• Graph and identify piecewise-defined functions, including open/closed circles and special values.

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

• Combine rational expressions and state all excluded values from denominators.

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

• 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 to find domain, range, intercepts, intervals, extrema, function values, and approximate solutions.

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

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

• Work with inverse values when a function’s domain is restricted.

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

• Expand logarithmic expressions using product, quotient, and power rules.

• Solve logarithmic equations and check domain restrictions before accepting a solution.

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

• Rewrite a linear equation in slope-intercept form and identify the slope and y-intercept.

• Write a polynomial for a shaded area by subtracting areas and simplifying.

• Set up a system for a quadratic model $y = ax^2 + bx + c$ from three data points.

• Choose an appropriate first factoring strategy: GCF, difference of squares, perfect square trinomial, grouping, trinomial by trial and error, or difference of cubes.