Calculus 1 - MTHG002 - Fall 2024

The absolute value of a number 'a' (|a|) is the distance from 'a' to zero on a number line.

Rules that govern how absolute values interact with each other and with operations like multiplication.

Finding the range of values for a variable 'x' that satisfy an inequality involving an absolute value.

Finding the values of 'x' that make a quadratic expression greater than or less than zero.

A function maps each input to a unique output; Domain is the set of all possible inputs.

A way to represent a range of numbers on a number line using brackets [ ] or parentheses ( ).

An inequality that involves a quadratic expression.

Find the range of values that satisfies an inequality with an absolute value.

A function created by adding terms that are multiples of powers of the variable 'x', where the powers are nonnegative integers.

Functions like f(x) = 2x - 1, g(x) = 3x² - x + 1, h(x) = 5x³ + x² - 2x + 7 are all polynomials since they follow the pattern of a polynomial function.

The domain of a polynomial function is the set of all real numbers (ℝ), meaning you can input any real number into the function and get an output.

A function formed by dividing one polynomial by another polynomial. It can be expressed as f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials.

The domain of a rational function excludes any values of 'x' that make the denominator equal to zero because division by zero is undefined..

A function created by combining polynomials using basic arithmetic operations (addition, subtraction, multiplication, division, and taking roots).

An algebraic function where the root is taken of a polynomial function. The general form is f(x) = n√g(x), where g(x) is a polynomial and n is a natural number.

The domain of a root function depends on the root's index (n): - If n is odd, the domain is all real numbers (ℝ). - If n is even, the domain is restricted to values of 'x' where the polynomial g(x) is greater than or equal to zero.

The set of all possible input values (x-values) for which a function is defined.

The set of all possible output values (y-values) that a function can produce.

A function is defined for all real values if it can take any real number as input without encountering any restrictions or undefined results.

Find the smallest output value (y-value) that the function can produce by finding the input value (x-value) that results in the smallest output.

Find the largest output value (y-value) that the function can produce by finding the input value (x-value) that results in the largest output.

A function defined by an expression under a square root symbol, where the expression inside the root cannot be negative.

Finding the range of values for a variable that satisfy an inequality.

A diagram used to visualize the signs (positive or negative) of expressions over different intervals.

The domain of an algebraic function is typically all real numbers except for values that make the denominator zero or introduce undefined operations.

Identify restrictions on the input values (x) of a function based on its definition, such as avoiding division by zero or square roots of negative numbers.

Certain values might be excluded from the domain to prevent undefined results like dividing by zero or taking the square root of a negative number.

This notation means 'x cannot be equal to -2'. It indicates that -2 is excluded from the function's domain.

This represents the domain of the function 'f'. The symbol 'ℝ' denotes all real numbers, and the set {-2, -1} is excluded from the domain.

This restriction prevents dividing by zero in the function 'g(x)'.

This inequality ensures that the expression inside the square root remains non-negative.

The domain of 'g(x)' includes all values less than or equal to -1 and greater than 2, expressed in interval notation.

This inequality guarantees that the expression within the square root is positive.

A vertical line that a function approaches but never touches as the input approaches a certain value. This happens typically when the denominator of a rational function equals zero.

A point on the graph of a function where the function is undefined, but the function can be made continuous by simplifying the expression. Often occurs when a factor cancels out in both numerator and denominator.

A transformation of a function's graph where the entire graph is moved left or right on the x-axis. This is achieved by adding or subtracting a constant to the input (x) within the function.

A transformation of a function's graph where the entire graph is moved up or down on the y-axis. This is achieved by adding or subtracting a constant outside the function.

A transformation of a function's graph where the graph is stretched vertically, making it taller or shorter. This is achieved by multiplying the function by a constant (between 0 and 1).

A technique used to rewrite a quadratic expression into a perfect square trinomial, usually to solve equations or find the vertex of a parabola. It involves manipulating the expression to create a perfect square form: (x + h)² + k.