Types of Mathematical Functions

Types of Mathematical Functions

Algebraic Functions

• A function that can be expressed as a polynomial or a rational function (i.e., the quotient of two polynomials)
• Examples: f(x) = x^2, f(x) = 1/x

Trigonometric Functions

• Functions that relate to the angles and triangles in a right triangle
• Examples: sin(x), cos(x), tan(x)

Exponential Functions

• Functions that have a base raised to a power
• Examples: f(x) = 2^x, f(x) = e^x

Logarithmic Functions

• Functions that are the inverse of exponential functions
• Examples: f(x) = log(x), f(x) = ln(x)

Function Operations

Composition

• Combining two functions by applying one function to the output of another
• Notation: (f ∘ g)(x) = f(g(x))

Inverse

• A function that reverses the effect of another function
• Notation: f^(-1)(x)

Function Properties

Domain

• The set of all input values for which the function is defined

Range

• The set of all possible output values of the function

Even and Odd Functions

• Even function: f(-x) = f(x)
• Odd function: f(-x) = -f(x)

Graphical Representations

Cartesian Coordinates

• A system of coordinates that uses x and y axes to plot points

Graph of a Function

• A visual representation of a function's behavior
• Can be used to identify key features such as maxima, minima, and asymptotes

Types of Mathematical Functions

• Algebraic functions can be expressed as a polynomial or a rational function.
• Examples of algebraic functions include f(x) = x^2 and f(x) = 1/x.
• Trigonometric functions relate to angles and triangles in a right triangle.
• Examples of trigonometric functions include sin(x), cos(x), and tan(x).
• Exponential functions have a base raised to a power.
• Examples of exponential functions include f(x) = 2^x and f(x) = e^x.
• Logarithmic functions are the inverse of exponential functions.
• Examples of logarithmic functions include f(x) = log(x) and f(x) = ln(x).

Function Operations

• Function composition combines two functions by applying one function to the output of another.
• Notation for function composition is (f ∘ g)(x) = f(g(x)).
• Inverse functions reverse the effect of another function.
• Notation for inverse functions is f^(-1)(x).

Function Properties

• Domain is the set of all input values for which the function is defined.
• Range is the set of all possible output values of the function.
• Even functions satisfy the condition f(-x) = f(x).
• Odd functions satisfy the condition f(-x) = -f(x).

Graphical Representations

• Cartesian coordinates use x and y axes to plot points.
• Graph of a function is a visual representation of a function's behavior.
• Graphs can be used to identify key features such as maxima, minima, and asymptotes.

Graphing Equations

• The general form of a linear equation is Ax + By = C, where A, B, and C are constants.
• To graph an equation, follow these steps: write the equation in slope-intercept form (y = mx + b), identify the slope (m) and y-intercept (b), plot the y-intercept on the coordinate plane, use the slope to find additional points on the line, and draw the line through the points.

Distance Formula

• The distance formula is used to calculate the distance between two points on a coordinate plane.
• The formula is: √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points.
• The distance formula can be used to find the length of a line segment, the distance between two points, or the radius of a circle.

Slope-intercept Form

• Slope-intercept form is a way of writing linear equations in the form y = mx + b.
• In slope-intercept form, m is the slope (a measure of how steep the line is) and b is the y-intercept (the point where the line crosses the y-axis).
• Slope-intercept form is useful for graphing equations and identifying the slope and y-intercept.

Midpoint Formula

• The midpoint formula is used to find the midpoint of a line segment.
• The formula is: ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment.
• The midpoint formula can be used to find the midpoint of a line segment, or to find the coordinates of the midpoint of a line segment given the endpoints.