Algebraic Foundations: Special Products, Factorization, and Functions

Special Products and Factorization Techniques

Quadratic Formula

The quadratic formula is used to solve quadratic equations of the form $ax^2 + bx + c = 0$.

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

• Example: Solve $x^2 + 3x - 1 = 0$ $x = \frac{-3 \pm \sqrt{13}}{2}$

Special Products

Special algebraic products simplify the process of expanding and factoring expressions.

• Square of a Binomial: $(x + a)^2 = x^2 + 2ax + a^2$ $(x - a)^2 = x^2 - 2ax + a^2$

• Product of Sum and Difference: $(x + a)(x - a) = x^2 - a^2$

• Cube of a Binomial: $(x + a)^3 = x^3 + 3x^2a + 3xa^2 + a^3$ $(x - a)^3 = x^3 - 3x^2a + 3xa^2 - a^3$

• Sum and Difference of Cubes: $x^3 + a^3 = (x + a)(x^2 - xa + a^2)$ $x^3 - a^3 = (x - a)(x^2 + xa + a^2)$

Binomial Theorem

The binomial theorem provides a formula for expanding powers of binomials:

• $(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k$

• Where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Factoring by Grouping

Factoring by grouping is a method used when a polynomial has four or more terms.

• Group terms to factor common factors, then factor out the common binomial.

• Example: $ax^3 + abx^2 + cx + cb = (ax^3 + abx^2) + (cx + cb) = a x^2(x + b) + c(x + b) = (x + b)(a x^2 + c)$

Functions

Notations and Definitions for a Cartesian Plane

The Cartesian plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points are represented as ordered pairs $(x, y)$.

• The point of intersection of the axes is called the origin $(0, 0)$.

• The axes divide the plane into four quadrants:

Quadrant	Sign of x	Sign of y
I	+	+
II	-	+
III	-	-
IV	+	-

Function and Notations

A function is a relation between two variables such that to each value of the independent variable, there corresponds exactly one value of the dependent variable.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Codomain: The set of all possible output values (y-values) that the function could possibly take.

• Range: The set of all actual output values (y-values) that the function takes for inputs from the domain.

• Functions can be represented by tables, graphs, or algebraic expressions.

Functions Specified by Algebraic Expressions

Not all equations in two variables define y as a function of x. For example:

• $y = x^2$ is a function (each x has one y).

• $x^2 + y^2 = 4$ is not a function (a single x can correspond to multiple y values).

• $x = y^2 + 1$ is not a function (fails the vertical line test).

Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Example: The graph of $y = x^2$ passes the vertical line test; $x^2 + y^2 = 4$ does not.

Domain and Range of Functions

To find the domain and range of a function, consider the following:

• For even roots (e.g., square roots), the expression under the root must be non-negative.

• For denominators, the denominator must not be zero.

• For logarithmic functions, the argument must be positive.

Examples:

• For $f(x) = \sqrt{x-5}$, domain: $x \geq 5$

• For $f(x) = \frac{1}{x-1}$, domain: $x \neq 1$

• For $f(x) = \sqrt{4-x^2}$, domain: $-2 \leq x \leq 2$

Even and Odd Functions

Even functions: Satisfy $f(-x) = f(x)$ for all $x$ in the domain. Their graphs are symmetric about the y-axis. Odd functions: Satisfy $f(-x) = -f(x)$ for all $x$ in the domain. Their graphs are symmetric about the origin. Neither: If neither condition is satisfied.

• Example (Even): $f(x) = x^2$

• Example (Odd): $f(x) = x^3$

• Example (Neither): $f(x) = x^2 + x$

Piecewise-Defined Functions

A piecewise-defined function uses different formulas for different parts of its domain. For example, the absolute value function:

• $f(x) = |x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

Increasing and Decreasing Functions

A function $f$ is increasing on an interval if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in the interval. It is decreasing if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.

Transformations of Functions

Transformations shift, stretch, compress, or reflect the graph of a function:

• $f(x) + c$: Shift up by $c$ units

• $f(x) - c$: Shift down by $c$ units

• $f(x + c)$: Shift left by $c$ units

• $f(x - c)$: Shift right by $c$ units

• $c f(x)$: Vertical stretch by $c$ ($c > 1$)

• $f(cx)$: Horizontal compression by $c$ ($c > 1$)

• $-f(x)$: Reflect about the x-axis

• $f(-x)$: Reflect about the y-axis

Operations on Functions

Functions can be added, subtracted, multiplied, or divided (except where the denominator is zero):

• $(f + g)(x) = f(x) + g(x)$

• $(f - g)(x) = f(x) - g(x)$

• $(f \cdot g)(x) = f(x)g(x)$

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

Composite Functions

The composite function $f \circ g$ is defined as $(f \circ g)(x) = f(g(x))$. The domain of $f \circ g$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

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

Exponential Functions

Definition and Properties

An exponential function is a function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Properties:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • y-intercept: $(0, 1)$

  • Horizontal asymptote: $y = 0$

  • Continuous and one-to-one

  • Always increasing if $a > 1$; always decreasing if $0 < a < 1$

Properties of Exponents

• $a^0 = 1$

• $a^m \cdot a^n = a^{m+n}$

• $\frac{a^m}{a^n} = a^{m-n}$

• $(a^m)^n = a^{mn}$

• $(ab)^n = a^n b^n$

• $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Natural Exponential Function

The natural exponential function uses the base $e \approx 2.71828$:

• $f(x) = e^x$

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

Graph Transformations for Exponential Functions

• Vertical shifts: $f(x) = a^x + c$

• Horizontal shifts: $f(x) = a^{x - h}$

• Reflections: $f(x) = -a^x$ (about x-axis)

• Stretch/compression: $f(x) = k a^x$

Summary Table: Exponential Function Characteristics

Function	Domain	Range	y-intercept	Asymptote
$f(x) = a^x$ ($a > 1$)	$(-\infty, \infty)$	$(0, \infty)$	$(0, 1)$	$y = 0$
$f(x) = e^x$	$(-\infty, \infty)$	$(0, \infty)$	$(0, 1)$	$y = 0$

Additional info: These algebraic and function concepts are foundational for further study in mathematics, including calculus and applications in sciences such as chemistry and physics.