Math 101.00 - Final Review

A mathematical expression that represents a set of numbers with an upper and lower bound. For example, (-3, 4] represents all numbers greater than -3 but less than or equal to 4. The notation (-3, 4] indicates that -3 is excluded from the set, while 4 is included.

The distance of a number from zero on the number line, always expressed as a positive value. For example, the absolute value of -5 is 5, written as |-5| = 5.

A numerical representation of a mathematical equation using a table. Each row in the table shows a specific value of the independent variable (x) and its corresponding dependent variable (y) as determined by the equation. For example, to create a table for y = x + 2, you would choose several x values and calculate the corresponding y values.

An equation that represents a straight line on a graph. It can be written in the form ax + by = c, where a, b, and c are constants.

A method of simplifying expressions that contain exponents by using the rules of exponents. These rules involve multiplying powers, dividing powers, raising a power to another power, and raising a product or quotient to a power. For example, x^3 * x^2 = x^(3+2) = x^5.

Scientific notation is a way to write very large or very small numbers conveniently using powers of ten. For example, 1,000 can be expressed in scientific notation as 1 × 10^3. Decimal notation expresses numbers in the standard format we are used to, with a decimal point separating the whole number part and the fractional part. For example, 1,000 is in decimal notation.

A set of ordered pairs where each input (x-value) has only one output (y-value). It means no two different ordered pairs have the same x-value.

The set of all possible input values (x-values) for a function. It represents the values that can be plugged into the function. For example, the domain of y = x^2 is all real numbers.

The set of all possible output values (y-values) for a function. It represents the range of values that the function can produce. For example, the range of y = x^2 is all non-negative real numbers.

A function where each input has exactly one output. It's a special type of relation that passes the vertical line test.

A specific value of a function for a given input. For example, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7. Here, f(3) is the function value when x is equal to 3.

A way to find the equation of a line using two points on the line. The formula is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

A representation of the relationship between two variables (x and y) in the form y = mx + c. Here, 'm' represents the slope, or the steepness of the line, and 'c' represents the y-intercept, or the point where the line crosses the y-axis.

A mathematical method to solve a system of two or more linear equations simultaneously. It involves rewriting one equation in terms of one variable and substituting it in the other equation.

A method for solving systems of linear equations. It's based on the idea of eliminating one variable by adding or subtracting the equations together. You manipulate the equations to get the same coefficient for one of the variables, then add or subtract the equations to eliminate it.

A mathematical statement showing that one quantity or expression is greater than (<), less than (>), greater than or equal to (>=), or less than or equal to (<=) another quantity or expression. For example, x < 5 means 'x is less than 5'.

An inequality that involves one variable. It's usually represented by an equation with an inequality symbol (<, >, <=, >=). For Example: x + 2 <= 5.

An inequality that includes both an 'and' or 'or' condition, specifying a solution set that satisfies both conditions (for AND) or at least one of the conditions (for OR).

A polynomial that has three terms. Its general form is ax^2 + bx + c, where 'a', 'b', and 'c' are constants. For example, 2x^2 + 3x - 1 is a trinomial.

A method of factoring quadratic expressions by grouping terms that share common factors. For example, 2x^2 + 4x + 3x + 6 can be factored by grouping: 2x(x + 2) + 3(x +2) = (2x + 3)(x + 2).

A mathematical expression that represents the ratio of two polynomials. For example, (x^2 + 1) / (x - 2) is a rational expression.

An equation that involves one or more rational expressions. To solve them, you typically multiply both sides of the equation by the least common multiple of the denominators to eliminate them. For example, (x + 1) / x = 2 can be solved by multiplying both sides by x.

A method of simplifying rational expressions by finding a common factor in the numerator and denominator and canceling it out. For example, (x^2 - 4) / (x + 2) can be simplified to x - 2 by factoring the numerator as (x + 2)(x - 2) and canceling the common factor (x + 2).

A way to divide one polynomial by another polynomial. It's similar to long division but using algebraic terms. For example, dividing (x^3 + 2x^2 + 1) by (x + 1) using polynomial long division.

A mathematical expression that involves the square root of a variable. It's written in the form √x, where 'x' is the radicand, the number under the radical sign.

Expressing a radical expression in a simplified form by removing any radicals from the denominator and simplifying the expression as much as possible. For example, √2 / √3 can be simplified by multiplying both numerator and denominator by √3, resulting in √6 / 3.

An equation that involves one or more radical expressions. To solve them, you typically isolate the radical on one side of the equation and then square both sides to eliminate the radical. For example, √(x + 2) = 3 can be solved by squaring both sides to eliminate the radical.

A mathematical expression that involves the square root of -1, denoted by 'i'. Complex numbers are typically written in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.

A method to solve quadratic equations by isolating the squared term on one side and then taking the square root of both sides. For example, (x - 2)^2 = 9 can be solved by taking the square root of both sides and solving for x.

A method to solve quadratic equations by manipulating it into a perfect square trinomial. You complete the square by adding a constant term to both sides of the equation to create a trinomial that can be factored as (x + a)^2 or (x - a)^2.

The shape of the graph of a quadratic function. It is a U-shaped curve, either opening upwards or downwards, depending on the sign of the coefficient of the x^2 term.

The lowest or highest point on a parabola, depending on whether it opens upwards or downwards. It is also the point where the parabola intersects the axis of symmetry.

A vertical line that divides a parabola into two symmetrical halves. It passes through the vertex of the parabola and is defined by the equation x = a, where 'a' is the x-coordinate of the vertex.

A function that increases exponentially, meaning its growth rate is proportional to its current value. The general form of an exponential function is f(x) = a*b^x, where 'a' is the initial value and 'b' is the growth factor.

A function that is the inverse of an exponential function. It's used to find the exponent to which a base must be raised to get a specific value. Its general form is f(x) = log_b(x).

Formulas that calculate the future value of an investment with compound interest. A = P(1 + r/n)^(nt) represents compound interest calculated n times per year, and A = Pe^(rt) represents continuous compounding. Here 'P' is the principal, 'r' is the annual interest rate, 'n' is the number of times interest is compounded per year, and 't' is the time in years.

A method of combining two functions by applying one function to the output of another function. It's denoted as (f ◦ g)(x), which means 'apply g to x and then apply f to the result'.

Two functions are inverses if their compositions cancel each other out. If f(g(x)) = x and g(f(x)) = x, then f and g are inverse functions.