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Honors Pre-Calculus
Table of Contents
Unit 1 Overview: Functions
1.1 Functions and Function Notation
1.2 Domain and Range
1.3 Rates of Change and Behavior of Graphs
1.4 Composition of Functions
1.5 Transformation of Functions
1.6 Absolute Value Functions
1.7 Inverse Functions

📏honors pre-calculus review

1.1 Functions and Function Notation

Citation:

Functions are the building blocks of math relationships. They show how inputs and outputs connect through equations, tables, graphs, or words. Understanding functions helps us model real-world situations and make predictions.

Functions can be one-to-one or many-to-one. We use tools like the vertical line test to check if a graph represents a function. Common function types include linear, quadratic, and exponential, each with unique shapes and properties.

Functions and Their Representations

Classification of function representations

  • Functions represented by equations express the relationship between input and output variables using mathematical symbols and operations
  • Functions represented by tables organize input and output values in columns or rows, showing the correspondence between them
  • Functions represented by graphs plot input values on the horizontal axis and output values on the vertical axis, visualizing the relationship
  • Functions represented by verbal descriptions explain the relationship between input and output using words, often in the context of a real-world situation

Interpretation of function outputs

  • Interpreting the output of a function involves understanding what the value represents in the given context
  • The units of the output should be considered when interpreting the result (dollars, meters, degrees)
  • In a real-world application, the output can provide meaningful information about the situation being modeled (profit, height, temperature)

One-to-one vs many-to-one functions

  • One-to-one functions have a unique output for each input, ensuring that no two input values map to the same output value
    • Example: $f(x) = 2x + 1$ is one-to-one because each input produces a unique output
  • Many-to-one functions can have multiple input values that map to the same output value
    • Example: $f(x) = x^2$ is many-to-one because both $x = 2$ and $x = -2$ map to the same output value of 4

Vertical line test for functions

  • The vertical line test determines if a relation (a set of ordered pairs) is a function based on its graphical representation
  • To apply the test, imagine drawing vertical lines through the graph
    • If any vertical line intersects the graph at more than one point, the relation is not a function
    • If no vertical line intersects the graph at more than one point, the relation is a function

Graphs of common functions

  • Linear functions have a constant rate of change and are represented by a straight line
    • The slope ($m$) represents the change in $y$ divided by the change in $x$
    • The $y$-intercept ($b$) is the $y$-value when $x = 0$
    • Example: $y = 2x + 3$ is a linear function with a slope of 2 and a $y$-intercept of 3
  • Quadratic functions have a parabolic shape and are represented by an equation in the form $y = ax^2 + bx + c$
    • The vertex is the maximum or minimum point of the parabola
    • The axis of symmetry is the vertical line that passes through the vertex
    • Example: $y = -x^2 + 4x - 3$ is a quadratic function with a vertex at $(2, 1)$
  • Exponential functions involve a constant base raised to a variable power and are represented by an equation in the form $y = a \cdot b^x$
    • $a$ is the initial value (the $y$-value when $x = 0$)
    • $b$ is the growth factor (if $b > 1$) or decay factor (if $0 < b < 1$)
    • Example: $y = 2 \cdot 3^x$ is an exponential growth function with an initial value of 2 and a growth factor of 3
  • Piecewise functions are defined by different equations for different intervals of the input variable

Advanced Function Concepts

  • Inverse functions reverse the relationship between input and output, effectively "undoing" the original function
  • Composition of functions involves applying one function to the output of another, creating a new function
  • Functions can have multiple inputs that together determine a single output

Key Terms to Review (26)

Y-intercept: The y-intercept is the point where a line or curve intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a critical parameter that describes the behavior of various functions, including linear, quadratic, polynomial, and exponential functions.
Piecewise Function: A piecewise function is a function that is defined by different formulas or expressions over different intervals or domains of the independent variable. These functions are composed of multiple parts, each with its own rule or equation, allowing for greater flexibility in modeling real-world phenomena.
Input: In the context of functions and function notation, input refers to the independent variable or the value that is fed into a function. It is the value that the function acts upon to produce a corresponding output or dependent variable.
Exponential Function: An exponential function is a mathematical function where the independent variable appears as the exponent. These functions exhibit a characteristic growth or decay pattern, with the value of the function increasing or decreasing at a rate that is proportional to the current value. Exponential functions are fundamental in understanding various real-world phenomena, from population growth to radioactive decay.
F(x): f(x) is a function notation that represents a relationship between an independent variable, x, and a dependent variable, f. The function f maps each input value of x to a unique output value of f(x), establishing a correspondence between the two variables.
Output: Output refers to the result or consequence of a function or process. It is the information, data, or value that is produced or generated as the end product of a mathematical, computational, or logical operation.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions describe how one quantity changes in relation to another, and they are widely used in various fields, including mathematics, science, and engineering.
Domain: The domain of a function refers to the set of all possible input values for the function. It represents the range of values that the independent variable can take on. The domain is a crucial concept in understanding the behavior and properties of various mathematical functions.
Range: The range of a function refers to the set of all possible output values or the set of all values that the function can attain. It represents the vertical extent or the interval of values that the function can produce as the input variable changes. The range is an important concept in the study of functions and their properties, as it provides information about the behavior and characteristics of the function.
Quadratic Function: A quadratic function is a polynomial function of degree two, meaning it contains a variable raised to the power of two. These functions are characterized by a parabolic shape and are widely used in various mathematical and scientific applications.
Inverse Function: An inverse function is a function that undoes the operation of another function. It is a special type of function that reverses the relationship between the input and output variables of the original function, allowing you to solve for the input when given the output.
Composition of Functions: Composition of functions is a mathematical operation where the output of one function becomes the input of another function. It allows for the combination of multiple functions to create a new function that performs a more complex task by chaining the individual functions together.
Many-to-One Function: A many-to-one function, also known as a surjective function, is a type of function where multiple elements in the domain (input set) can map to the same element in the codomain (output set). In other words, a single output value can correspond to multiple input values, but each input value is associated with only one output value.
Growth Factor: A growth factor is a naturally occurring substance capable of stimulating cellular growth, proliferation, healing, and differentiation. Growth factors are crucial in the regulation of a variety of cellular processes, including cell growth, cell differentiation, cell migration, and cell survival, and have applications in both the development of tissues and the progression of disease.
X-Intercept: The x-intercept of a graph is the point where the graph intersects the x-axis, indicating the value of x when the function's output or y-value is zero. It represents the horizontal location where a curve or line crosses the x-axis.
Y = f(x): The expression 'y = f(x)' represents a function, where 'y' is the dependent variable and 'x' is the independent variable. This notation indicates that the value of 'y' is determined by the value of 'x' through the function 'f'. The function 'f' is a rule or a relationship that maps each input value of 'x' to a unique output value of 'y'.
Vertex: The vertex is a key point on a graph or function that represents the maximum or minimum value of the function. It is the point where the graph changes direction, either from increasing to decreasing or vice versa.
Vertical Line Test: The vertical line test is a graphical method used to determine whether a relation or function is a function. It involves drawing a vertical line through the graph and checking if the line intersects the graph at only one point for each value of the independent variable.
Relation: A relation is a set of ordered pairs that connects elements from one set (the domain) to elements in another set (the range). Relations are a fundamental concept in mathematics, particularly in the study of functions and their properties.
Codomain: The codomain of a function is the set of all possible output values that the function can produce. It represents the full range of values that the function is capable of mapping its input values to.
One-to-One Function: A one-to-one function, also known as an injective function, is a special type of function where each element in the domain is paired with a unique element in the codomain. In other words, for any two distinct elements in the domain, their corresponding elements in the codomain must also be distinct.
Decay Factor: The decay factor is a measure of how quickly a quantity or function decreases over time. It is a crucial concept in the study of exponential functions and the analysis of data that exhibits exponential behavior, such as radioactive decay or population growth.
Decreasing: Decreasing refers to a quantity or function that becomes smaller or lower in value over time or as the independent variable increases. It is the opposite of increasing, where a quantity or function becomes larger or higher in value.
Axis of Symmetry: The axis of symmetry is a line that divides a symmetric figure or function into two equal and mirror-like halves. It is a critical concept in understanding the properties and behavior of various mathematical functions and geometric shapes.
Slope: Slope is a measure of the steepness or incline of a line. It represents the rate of change in the vertical direction (y-coordinate) compared to the change in the horizontal direction (x-coordinate) along a line. Slope is a fundamental concept in understanding the properties and behaviors of linear functions.
Increasing: Increasing refers to a function where the output values rise as the input values rise. This characteristic indicates that for any two points on the graph of the function, if one point has a smaller x-value than the other, the corresponding y-value of the first point will be less than that of the second. Understanding this concept is crucial as it connects to the overall behavior of functions and their graphs, helping to identify trends and changes in data.