College Algebra Review: Functions, Transformations, Equations, and Inequalities

Functions and Their Properties

Increasing, Decreasing, and Constant Functions

Understanding where a function is increasing, decreasing, or constant is fundamental in analyzing its behavior.

• Increasing: A function f(x) is increasing on an interval if, for any two numbers a and b in the interval with a < b, f(a) < f(b).

• Decreasing: A function is decreasing on an interval if, for any a < b, f(a) > f(b).

• Constant: A function is constant on an interval if f(a) = f(b) for all a and b in the interval.

• Example: Given a graph, identify intervals where the function rises (increasing), falls (decreasing), or remains flat (constant).

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Definition: A piecewise function is a function composed of multiple sub-functions, each applying to a certain interval of the main function's domain.

• Example: For $f(x) = \begin{cases} 2x & \text{if } x < -1 \\ -2x + 3 & \text{if } -1 \leq x \leq 3 \\ \sqrt{x} & \text{if } x > 3 \end{cases}$

  • To find f(-3), f(1), and f(9), substitute each value into the appropriate piece.

Operations with Functions

Function Addition, Subtraction, Multiplication, and Division

Functions can be combined using arithmetic operations, and the domain of the resulting function is the intersection of the domains of the original functions (except for division, where the denominator cannot be zero).

• Addition: $(f+g)(x) = f(x) + g(x)$

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

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

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

• Example: If $f(x) = x^2 + 3x - 4$ and $g(x) = 5x - 3$, then $(f+g)(x) = x^2 + 8x - 7$.

• Domain: The set of all $x$ for which the operation is defined.

Composition of Functions

The composition of functions involves applying one function to the result of another.

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

• Example: If $f(x) = \frac{1}{x}$ and $g(x) = \frac{x+1}{x-1}$, then $(f \circ g)(x) = f\left(\frac{x+1}{x-1}\right) = \frac{1}{\frac{x+1}{x-1}} = \frac{x-1}{x+1}$

• Domain: All $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Transformations of Functions

Types of Transformations

Transformations change the position or shape of a graph. Common transformations include translations, reflections, stretches, and compressions.

• Vertical Shift: $f(x) + k$ shifts the graph up by $k$ units if $k > 0$, down if $k < 0$.

• Horizontal Shift: $f(x - h)$ shifts the graph right by $h$ units if $h > 0$, left if $h < 0$.

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

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

• Example: For $f(x) = x^2$, $g(x) = -2(x+4)^2 + 4$ is reflected over the $x$-axis, vertically stretched by 2, shifted left 4 units, and up 4 units.

Writing Equations for Transformed Graphs

• Given a transformation, write the new equation by applying the appropriate changes to the parent function.

• Example: Stretch $f(x) = x^2$ vertically by 3, shift up 3, and left 4: $g(x) = 3(x+4)^2 + 3$

Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers of the form $a + bi$, where $i = \sqrt{-1}$.

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Example: $(2 - 7i) + (3 + 4i) = 5 - 3i$

Quadratic Equations and Their Properties

Solving Quadratic Equations

• Completing the Square: Rewrite $ax^2 + bx + c = 0$ in the form $(x + d)^2 = e$ and solve for $x$.

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

• Example: Solve $x^2 + 8x - 9 = 0$ by completing the square.

Vertex Form and Graphing

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Vertex: The point $(h, k)$

• Axis of Symmetry: $x = h$

• Minimum/Maximum: If $a > 0$, the parabola opens up (minimum); if $a < 0$, opens down (maximum).

• Example: Express $f(x) = x^2 + 4x - 1$ in vertex form, find the vertex, axis of symmetry, and minimum value.

Solving Equations and Inequalities

Rational and Radical Equations

• Rational Equations: Equations involving fractions with polynomials in the numerator and denominator.

• Example: $\frac{2}{x+5} + \frac{1}{x-5} = \frac{16}{x^2-25}$

• Radical Equations: Equations involving roots, such as $\sqrt{x+1} = x - 1$.

• Solving: Isolate the radical, square both sides, and check for extraneous solutions.

Absolute Value and Rational Inequalities

• Absolute Value Inequality: $|2x + 5| < 9$

• Solution: $-9 < 2x + 5 < 9$; solve for $x$ and express in interval notation.

• Rational Inequality: $\frac{4x+1}{x+2} \leq 3$

• Solution: Bring all terms to one side, find critical points, test intervals, and express the solution in interval notation.

Polynomials and End Behavior

End Behavior of Polynomials

The end behavior of a polynomial describes how the function behaves as $x$ approaches $\infty$ or $-\infty$.

• Leading Term Test: The term with the highest degree determines the end behavior.

• Example: For $P(x) = -2x^4 + 3x^2 - 2x + 4$, as $x \to \infty$, $P(x) \to -\infty$; as $x \to -\infty$, $P(x) \to -\infty$.

Summary Table: Function Transformations

Transformation	Equation	Effect on Graph
Vertical Shift	$f(x) + k$	Up by $k$ units if $k > 0$, down if $k < 0$
Horizontal Shift	$f(x - h)$	Right by $h$ units if $h > 0$, left if $h < 0$
Vertical Stretch/Compression	$a f(x)$	Stretched if $|a| > 1$, compressed if $0 < |a| < 1$
Reflection over $x$-axis	$-f(x)$	Flips graph over $x$-axis
Reflection over $y$-axis	$f(-x)$	Flips graph over $y$-axis

Additional info:

• Some questions reference specific sections (e.g., Section 2.1, 2.2, etc.), which likely correspond to a standard College Algebra textbook.

• All topics are core to a College Algebra course, including function analysis, transformations, complex numbers, solving equations and inequalities, and polynomial behavior.