Relations and Functions

Relations

• A relation is a set of ordered pairs.
• Relations can be represented as a set of ordered pairs, a table, a mapping diagram, a graph, or an equation.
• The domain of a relation is the set of all first elements (x-values) in the ordered pairs.
• The range of a relation is the set of all second elements (y-values) in the ordered pairs.

Functions

• A function is a special type of relation where each element of the domain (x-value) is paired with exactly one element of the range (y-value).
• In simpler terms, for every input, there is only one output.
• The vertical line test is used to determine if a graph represents a function: a vertical line should intersect the graph at most once.
• If a vertical line can be drawn that intersects the graph more than once, the relation is not a function.

Real World Graphs

• Real-world graphs are used to visually represent relationships between two variables in practical scenarios.
• The independent variable (x-axis) influences the dependent variable (y-axis).
• Understanding the axes and their units is crucial for interpreting the graph.
• Analyzing the slope and intercepts can provide insights into the relationship.
• The shape of the graph can indicate whether the relationship is linear, exponential, or some other type.

Equations as Functions

• An equation can represent a function if it expresses y as a function of x, meaning for each x, there is only one y.
• To determine if an equation represents a function, solve the equation for y.
• If solving for y results in a single expression without any ± signs, then the equation represents a function.
• Equations like y = mx + b (linear equations) typically represent functions.
• Equations like x² + y² = r² (circles) do not represent functions because solving for y would involve a ± sign.

Graphing Functions using Tables

• A table can be used to graph a function by choosing values for x, substituting them into the function's equation to find the corresponding y-values, and then plotting the resulting ordered pairs (x, y).
• Choose a variety of x-values, including positive, negative, and zero, to get a good representation of the function's graph.
• Plot the points from the table on a coordinate plane.
• Connect the points with a smooth curve or line to create the function's graph.

Function Notation

• Function notation is a way of writing functions that makes it easy to see the input and output.
• It generally looks like f(x), where 'f' is the name of the function, and 'x' is the input variable.
• f(x) is read as "f of x," and it represents the value of the function at x.
• For example, if f(x) = x² + 3, then f(2) means substitute x = 2 into the function: f(2) = 2² + 3 = 7.
• Function notation helps to clearly indicate the input value and the corresponding output value.

Finding Zeros Graphically

• Zeros of a function are the x-values for which the function's output (y-value) is zero.
• Graphically, zeros are the points where the graph of the function intersects the x-axis.
• To find zeros graphically, plot the graph of the function and identify the x-coordinates of the x-intercepts.
• These x-coordinates are the zeros (or roots) of the function.

Finding Zeros Algebraically

• To find the zeros of a function algebraically, set the function equal to zero and solve for x.
• For example, if f(x) = 2x - 6, set 2x - 6 = 0 and solve for x: 2x = 6, so x = 3. The zero of the function is x = 3.
• For quadratic functions, factoring, completing the square, or using the quadratic formula may be necessary to find the zeros.
• The number of zeros depends on the type of function. Linear functions have at most one zero, quadratic functions have at most two zeros, etc.

Arithmetic Sequences

• An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.
• This constant difference is called the common difference, denoted by 'd'.
• Arithmetic sequences can be defined recursively or explicitly.
• In a recursive definition, each term is defined in relation to the previous term: aₙ = aₙ₋₁ + d, where a₁ is the first term.
• In an explicit definition, each term is defined directly in terms of 'n': aₙ = a₁ + (n - 1)d, where a₁ is the first term, 'n' is the term number, and 'd' is the common difference.
• Given any two terms in the sequence one can find the explicit formula for said sequence
• To find the sum of the first 'n' terms of an arithmetic sequence use the formula: Sₙ = n/2(a₁ + aₙ).