Graphs, Functions, and Systems: Study Guide for College Algebra

Graphs and Functions

Intervals of Continuity

Understanding where a function is continuous is essential for analyzing its behavior. A function is continuous on an interval if its graph can be drawn without lifting your pencil.

• Continuous Interval: The set of all x-values where the function has no breaks, holes, or jumps.

• Discontinuity: Points where the function is not defined or has a sudden change in value.

• Example: The function $f(x) = \frac{1}{x}$ is continuous everywhere except at $x = 0$.

Basic Functions and Their Graphs

Recognizing the shapes of basic functions helps in graphing and understanding more complex functions.

• Linear Function: $f(x) = x$ (straight line through the origin)

• Quadratic Function: $f(x) = x^2$ (parabola opening upwards)

• Absolute Value Function: $f(x) = |x|$ (V-shaped graph)

• Greatest Integer Function: $f(x) = \lfloor x \rfloor$ (step function)

• Example: The graph of $f(x) = x^3$ is an S-shaped curve passing through the origin.

Piecewise-Defined Functions

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

• Graphing: Plot each piece on its specified interval.

• Evaluating: Determine which piece applies for the given input value.

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

Greatest Integer Function

The greatest integer function (also called the floor function) returns the largest integer less than or equal to x.

• Notation: $f(x) = \lfloor x \rfloor$

• Example: $\lfloor 2.7 \rfloor = 2$, $\lfloor -1.3 \rfloor = -2$

Graphing Techniques (Transformations)

Functions can be shifted, reflected, stretched, or compressed using transformations.

• Vertical Shift: $f(x) + k$ shifts up by $k$ units.

• Horizontal Shift: $f(x - h)$ shifts right by $h$ units.

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

• Example: $f(x) = (x - 2)^2 + 3$ is $f(x) = x^2$ shifted right 2 units and up 3 units.

Symmetry of Graphs

Testing for symmetry helps classify functions and predict their graphs.

• x-axis Symmetry: Replace $y$ with $-y$; if unchanged, symmetric about x-axis.

• y-axis Symmetry: Replace $x$ with $-x$; if unchanged, symmetric about y-axis.

• Origin Symmetry: Replace $x$ with $-x$ and $y$ with $-y$; if unchanged, symmetric about the origin.

• Example: $f(x) = x^2$ is symmetric about the y-axis.

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 (y-axis symmetry).

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

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

Arithmetic Operations on Functions

Functions can be added, subtracted, multiplied, or divided (where defined).

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

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

Combinations and Compositions of Functions

Combining functions involves arithmetic operations or composition (using one function as the input of another).

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

• Domain: The set of all $x$ such that $x$ is in the domain of $g$ and $g(x)$ is in the domain of $f$.

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

Difference Quotient

The difference quotient is used to find the average rate of change of a function and is foundational in calculus.

• Formula: $\frac{f(x + h) - f(x)}{h}$, $h \neq 0$

• Example: For $f(x) = x^2$, the difference quotient is $\frac{(x + h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Systems of Equations and Inequalities

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solutions are points that satisfy all equations simultaneously.

• Graphical Method: Plot each equation; the intersection point(s) are the solutions.

• Substitution Method: Solve one equation for a variable and substitute into the other.

• Elimination (Addition) Method: Add or subtract equations to eliminate a variable, then solve for the remaining variable.

• Example: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ by adding to get $2x = 6 \Rightarrow x = 3$, then $y = 2$.

Application Problems Involving Systems

Systems of equations can model real-world problems, such as mixtures, investments, or supply and demand.

• Set up equations based on the problem's conditions.

• Solve using substitution, elimination, or graphing.

• Example: If two numbers add to 10 and their difference is 4, set up $x + y = 10$, $x - y = 4$.

Systems of Linear Equations in Three Variables

Systems with three variables can be solved using elimination or substitution, often resulting in a unique solution, infinitely many solutions, or no solution.

• Write the system: $\begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases}$

• Eliminate variables step by step to reduce to two equations in two variables, then solve.

• Example: $\begin{cases} x + y + z = 6 \\ x - y + z = 2 \\ x + y - z = 0 \end{cases}$

Solving Linear and Non-Linear Inequalities and Systems

Inequalities can be solved algebraically or graphically. Systems of inequalities define regions in the plane.

• Linear Inequality: $ax + by \leq c$

• Non-Linear Inequality: Involves powers or products of variables, e.g., $x^2 + y^2 \leq 1$

• Graphing: Shade the region that satisfies all inequalities.

• Example: The solution to $y > 2x + 1$ is the region above the line $y = 2x + 1$.

Summary Table: Key Concepts and Methods

Topic	Key Points	Example
Piecewise Function	Defined by different rules for different intervals	$f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases}$
Greatest Integer Function	Returns largest integer ≤ x	$\lfloor 3.7 \rfloor = 3$
Function Operations	Add, subtract, multiply, divide functions	$(f + g)(x) = f(x) + g(x)$
Composition	Apply one function to the result of another	$(f \circ g)(x) = f(g(x))$
Difference Quotient	Measures average rate of change	$\frac{f(x + h) - f(x)}{h}$
System of Equations	Solve by graphing, substitution, elimination	$\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$
System of Inequalities	Find region satisfying all inequalities	$y > x$, $y < 2x + 1$

Additional info: This guide covers material from sections 2.6, 2.7, 2.8, 5.1, and 5.6, as referenced in the assignments. All explanations are expanded for clarity and exam preparation.