Glossary

1.1 Review of Functions

5 min read•june 24, 2024

Functions are the building blocks of calculus. They describe relationships between variables, allowing us to model real-world phenomena. Understanding , types, and evaluation is crucial for solving complex problems in calculus.

and define where functions operate. Graphing functions reveals their behavior, including key features like intercepts and . Identifying helps solve equations and understand behavior, setting the stage for more advanced calculus concepts.

Function Fundamentals

Function notation evaluation

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  • Function notation [f(x)](https://www.fiveableKeyTerm:f(x))[f(x)](https://www.fiveableKeyTerm:f(x)) represents the output value when the is xx
    • Evaluate f(a)f(a) by substituting aa for xx in the function's equation and simplify
  • Function types and their notation
    • f(x)=mx+bf(x) = mx + b with slope mm and bb
    • f(x)=ax2+bx+cf(x) = ax^2 + bx + c with constants aa, bb, and cc
    • f(x)=anxn+an−1xn−1+…+a1x+a0f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 with constants aia_i
    • f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)} with polynomial functions P(x)P(x) and Q(x)Q(x)
    • f(x)=axf(x) = a^x with positive constant aa
    • f(x)=log⁡a(x)f(x) = \log_a(x) with positive constant a≠1a \neq 1
    • sin⁡(x)\sin(x), cos⁡(x)\cos(x), tan⁡(x)\tan(x), etc.
    • defined by different equations over different intervals of the

Domain and range identification

  • Domain: Set of all possible input values (usually xx) for a function
    • Most functions have a domain of all real numbers, unless restricted
      • Rational functions exclude xx values that make the denominator zero
      • Square root functions require a non-negative argument under the square root
      • Logarithmic functions require a positive argument
  • : Set of all possible output values (usually yy) for a function
    • Determined by the function's equation and domain restrictions
      • Linear functions have a range of all real numbers
      • Quadratic functions have a range depending on the parabola direction (up or down)
      • Exponential functions always have a positive range
      • Logarithmic functions have a range of all real numbers
      • Trigonometric functions have a limited range (e.g., −1≤sin⁡(x)≤1-1 \leq \sin(x) \leq 1)

Function graphing and features

  • Key features of function graphs
    • (zeros) where the graph crosses the x-axis (y=0y = 0)
    • y-intercept where the graph crosses the y-axis (x=0x = 0)
    • across the y-axis (), origin (), or other lines
    • Asymptotes that the graph approaches as xx or yy approaches infinity or a specific value
      • occur at xx values where the function is undefined (e.g., denominators equal to zero)
      • occur when the function approaches a constant value as xx approaches positive or negative infinity
    • where the function is increasing or decreasing
    • where the function reaches a highest or lowest value within a specific interval
    • indicating the direction of the curve (upward or downward)
    • where the concavity changes

Zeros of functions

  • Zeros () of a function are the xx values where f(x)=0f(x) = 0
    • Algebraic methods
      1. : Factor the function and set each factor equal to zero
      2. : For quadratic functions, use x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
      3. : For polynomial functions, list potential rational zeros and test using substitution
    • Graphical methods
      • Identify the x-intercepts of the function's graph

Function Representations and Operations

Function representations

  • Tables list input (xx) and output (yy or f(x)f(x)) values in two columns
    • Identify patterns in the table to determine the function type
  • Graphs plot points (xx, yy) on a coordinate plane and connect them to create a curve or line
    • Analyze the graph's key features to identify the function type
  • Equations express the relationship between the input and output using mathematical symbols
    • Identify the function type based on the equation's form (linear, quadratic, exponential)

Function operations and composition

  • Arithmetic operations on functions
    • Addition (f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x)
    • Subtraction (f−g)(x)=f(x)−g(x)(f - g)(x) = f(x) - g(x)
    • Multiplication (f⋅g)(x)=f(x)⋅g(x)(f \cdot g)(x) = f(x) \cdot g(x)
    • Division (fg)(x)=f(x)g(x)(\frac{f}{g})(x) = \frac{f(x)}{g(x)} where g(x)≠0g(x) \neq 0
  • of functions combines two or more functions by using the output of one as the input of another
    • (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)): First, evaluate g(x)g(x), then use the result as the input for ff
    • Domain of the is the domain of gg restricted to values that produce outputs within the domain of ff

Symmetry in functions

  • Even functions are symmetric about the y-axis
    • Definition: f(−x)=f(x)f(-x) = f(x) for all xx in the domain
    • Examples: f(x)=x2f(x) = x^2, f(x)=cos⁡(x)f(x) = \cos(x)
  • Odd functions are symmetric about the origin
    • Definition: f(−x)=−f(x)f(-x) = -f(x) for all xx in the domain
    • Examples: f(x)=x3f(x) = x^3, f(x)=sin⁡(x)f(x) = \sin(x)
  • Functions that are neither even nor odd have no symmetry or are symmetric about other lines (e.g., y=xy = x)
    • Example: f(x)=x2+xf(x) = x^2 + x
  • Implications on graphs
    • Even functions: If (a,b)(a, b) is on the graph, then (−a,b)(-a, b) is also on the graph
    • Odd functions: If (a,b)(a, b) is on the graph, then (−a,−b)(-a, -b) is also on the graph

Function Transformations and Continuity

  • alter the
    • Vertical shifts: f(x)+kf(x) + k moves the graph up kk units
    • Horizontal shifts: f(x−h)f(x - h) moves the graph right hh units
    • Vertical stretches/compressions: af(x)af(x) stretches (∣a∣>1|a| > 1) or compresses (0<∣a∣<10 < |a| < 1) vertically
    • Horizontal stretches/compressions: f(bx)f(bx) stretches (0<∣b∣<10 < |b| < 1) or compresses (∣b∣>1|b| > 1) horizontally
    • Reflections: −f(x)-f(x) reflects over the x-axis, f(−x)f(-x) reflects over the y-axis
  • have no breaks, holes, or jumps in their graphs
    • A function f(x)f(x) is continuous at a point aa if:
      1. f(a)f(a) is defined
      2. lim⁡x→af(x)\lim_{x \to a} f(x) exists
      3. lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)
    • A function is continuous on an interval if it is continuous at every point in that interval
  • have at least one point where continuity conditions are not met
    • Types of discontinuities:
      • Removable (point) discontinuity: A hole in the graph
      • Jump discontinuity: The function "jumps" from one value to another
      • Infinite discontinuity: The function approaches infinity as x approaches a certain value
  • "undo" the original function, swapping input and output
    • For a function ff, its inverse f−1f^{-1} satisfies f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x
    • The graph of an inverse function is the reflection of the original function over the line y=xy = x

Key Terms to Review (54)

Asymptotes: Asymptotes are imaginary lines that a curve approaches but never touches. They provide important information about the behavior and properties of a function, especially in the context of analyzing the function's behavior as it approaches certain values or as the independent variable approaches certain values.
Composite function: A composite function is a function created by applying one function to the results of another. It is denoted as $(f \circ g)(x) = f(g(x))$.
Composition: Composition is the act of combining or arranging multiple elements, functions, or operations into a unified whole. It is a fundamental concept in mathematics and various fields, describing how different components interact and integrate to form a cohesive structure or process.
Concavity: Concavity refers to the direction in which a curve bends, indicating whether it is curving upwards or downwards. A function is concave up if its graph opens upwards like a cup, meaning that its second derivative is positive, while it is concave down if the graph opens downwards, indicating a negative second derivative. Understanding concavity is essential for analyzing the behavior of functions, particularly when it comes to identifying intervals of increase and decrease as well as determining the nature of critical points.
Continuous Functions: Continuous functions are mathematical functions that have no abrupt changes or jumps in their values as the input variable changes. They represent a smooth, unbroken curve where the function value changes gradually and predictably as the input is varied.
Dependent variable: A dependent variable is a variable whose value depends on one or more other variables. In functions, it is typically represented as $y$ in the equation $y = f(x)$.
Discontinuous Functions: A discontinuous function is a function that is not defined at one or more points within its domain. This means the function has a jump, break, or gap in its graph, preventing it from being continuous throughout its entire range.
Domain: The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. It represents all the values that can be plugged into the function without causing any undefined behavior.
Domain: The domain of a function refers to the set of all possible input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is a crucial concept in understanding the behavior and properties of functions.
Endpoints: Endpoints are the points that mark the boundaries of an interval on a number line or graph. They can be either included in the interval (closed) or excluded (open).
Even function: An even function is a function $f(x)$ that satisfies the condition $f(-x) = f(x)$ for all $x$ in its domain. Graphically, even functions are symmetric with respect to the y-axis.
Even Functions: An even function is a function where the value of the function is the same for inputs that are equidistant from the origin, but on opposite sides. In other words, for any input x, the function value f(x) is equal to the function value f(-x).
Exponential Functions: Exponential functions are a class of mathematical functions where the independent variable appears as the exponent. These functions exhibit rapid growth or decay and are characterized by a constant rate of change, making them an important concept in calculus and various scientific fields.
F(x): f(x) is a mathematical function that represents a relationship between an independent variable x and a dependent variable y. It is a fundamental concept in calculus that describes how a quantity varies with respect to changes in another quantity.
Factoring: Factoring is the process of breaking down a mathematical expression, such as a polynomial or an algebraic expression, into a product of simpler factors. This technique is essential in various mathematical contexts, including the analysis of functions, limits, and asymptotes.
Function: A function is a relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output. It can be represented mathematically as $f(x)$, where $x$ is the input variable.
Function Notation: Function notation is a way of representing a function using a symbolic expression that clearly identifies the independent and dependent variables. It provides a concise and standardized method for expressing the relationship between the input and output values of a function.
Function Operations: Function operations refer to the mathematical manipulations and transformations that can be performed on functions to create new functions. These operations allow for the combination, modification, and analysis of functions, enabling the exploration of more complex relationships and behaviors.
Function Representations: Function representations refer to the various ways in which functions can be expressed or depicted, including algebraic formulas, graphs, tables, and verbal descriptions. These representations provide different perspectives on the properties and behavior of functions, allowing for a more comprehensive understanding of their characteristics.
Function Transformations: Function transformations refer to the process of modifying the graph of a function by applying various operations, such as translations, reflections, stretches, and compressions. These transformations allow for the creation of new functions from existing ones, providing a powerful tool for analyzing and understanding the behavior of functions.
Graph of a function: A graph of a function is a visual representation of all the ordered pairs $(x, y)$ that satisfy the function $y = f(x)$. It helps in understanding the behavior and properties of the function.
Horizontal Asymptotes: A horizontal asymptote is a horizontal line that a function's graph approaches as the input variable (typically x) approaches positive or negative infinity. It represents the limiting value that the function approaches but never actually reaches.
Independent variable: An independent variable is a variable that represents the input or cause and is not affected by other variables in the function. It is typically plotted on the horizontal axis of a graph.
Inflection Points: Inflection points are points on a curve where the curve changes from being concave up to concave down, or vice versa. They represent a critical transition in the behavior of a function, marking a shift in the direction of the curve's curvature.
Input: An input is the value or set of values that are fed into a function. It is often represented by the variable $x$ in equations and graphs.
Interval notation: Interval notation is a mathematical notation used to describe sets of numbers lying within a specific range. It uses brackets and parentheses to denote inclusive and exclusive bounds, respectively.
Intervals of Increase/Decrease: Intervals of increase and decrease refer to the regions within the domain of a function where the function's value is either increasing or decreasing. These intervals are crucial in understanding the behavior and characteristics of a function, as they provide insights into the function's overall shape and trends.
Inverse Functions: An inverse function is a function that undoes the operation of another function. It reverses the relationship between the input and output variables, allowing the output of the original function to become the input of the inverse function, and vice versa.
Linear Functions: A linear function is a type of function where the relationship between the independent variable (input) and the dependent variable (output) is represented by a straight line. This means that the change in the output variable is proportional to the change in the input variable.
Local Maxima/Minima: Local maxima and local minima refer to the points on a function where the function's value is greater than or less than the values in the immediate vicinity, respectively. These points represent the local highest and lowest points of the function within a specific region, as opposed to the overall maximum or minimum of the function over its entire domain.
Logarithmic Functions: Logarithmic functions are a class of functions that describe the relationship between two quantities, where one quantity is the exponent that a fixed base must be raised to in order to get the other quantity. They are the inverse functions of exponential functions and have important applications in various fields, including mathematics, science, and engineering.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. The graph of an odd function is symmetric about the origin.
Odd Functions: An odd function is a function that satisfies the condition $f(-x) = -f(x)$ for all $x$ in the function's domain. This means that the graph of an odd function is symmetric about the origin, with the graph reflecting across both the $x$-axis and the $y$-axis.
Piecewise Functions: A piecewise function is a mathematical function that is defined by multiple sub-functions, each valid for a different interval or domain of the independent variable. These sub-functions are stitched together to form the complete function, allowing it to exhibit different behaviors or characteristics across different regions of its domain.
Piecewise-defined functions: A piecewise-defined function is a function composed of multiple sub-functions, each defined on a specific interval of the domain. The overall function's definition changes depending on the input value.
Polynomial Functions: A polynomial function is a function that can be expressed as the sum of a finite number of terms, each of which is the product of a constant and one or more variables raised to a non-negative integer power. Polynomial functions are a fundamental class of functions in mathematics and are widely used in various fields, including science, engineering, and economics.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are second-degree polynomial equations in the form of $ax^2 + bx + c = 0$. It provides a systematic way to find the roots or solutions of a quadratic equation.
Quadratic Functions: Quadratic functions are a type of polynomial function where the highest exponent of the independent variable is 2. They are characterized by a U-shaped graph and can be used to model a variety of real-world phenomena, such as the motion of projectiles, the growth of populations, and the optimization of economic processes.
Range: The range of a function is the set of all possible output values (y-values) that the function can produce. It is determined by evaluating the function over its domain.
Range: Range refers to the set of all possible output values (or 'y' values) that a function can produce based on its domain (the set of input values). Understanding the range helps us grasp how a function behaves, what outputs are attainable, and the limitations on those outputs.
Rational Functions: A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a mathematical expression that can be used to model a wide range of real-world phenomena, from population growth to the behavior of electrical circuits.
Rational Root Theorem: The Rational Root Theorem is a mathematical principle that provides a method for finding the possible rational roots of a polynomial equation. It states that if a polynomial equation with integer coefficients has a rational root, then that root must be a factor of the constant term of the equation.
Roots: In the context of functions, the roots of a function refer to the values of the independent variable (usually denoted as x) where the function equals zero. Roots are the points where the graph of the function intersects the x-axis, indicating the solutions to the equation f(x) = 0.
Symmetry: Symmetry refers to the balanced and proportional arrangement of elements in a design or object. It describes the quality of being made up of exactly similar parts facing each other or around an axis, center, or edge.
Symmetry about the origin: A function is symmetric about the origin if rotating its graph 180 degrees around the origin does not change the graph. Mathematically, this means $f(-x) = -f(x)$ for all $x$ in the domain of $f$.
Symmetry about the y-axis: A function is symmetric about the y-axis if for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph. This implies that $f(x) = f(-x)$ for all x in the domain of the function.
Table of values: A table of values lists pairs of numbers that satisfy a given function. It is used to plot the graph of the function by showing specific input-output relationships.
Trigonometric Functions: Trigonometric functions are mathematical functions that describe the relationships between the sides and angles of a right triangle. They are widely used in various fields, including calculus, to analyze periodic phenomena and model real-world situations.
Vertical Asymptotes: A vertical asymptote is a vertical line that a function's graph approaches but never touches. It represents the value of the independent variable where the function is undefined or has a discontinuity.
Vertical line test: The vertical line test is a method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
X-Intercepts: The x-intercepts of a function are the points where the graph of the function intersects the x-axis, or the values of x where the function equals zero. They represent the horizontal coordinates of the points where the function crosses the x-axis.
Y-intercept: The y-intercept is the point at which a graph or function intersects the y-axis, representing the value of the function when the input (x-value) is zero. It is a crucial concept in understanding the behavior and properties of various functions, including linear, exponential, and logarithmic functions.
Zeros: Zeros, in the context of mathematical functions, refer to the points where the function's value is equal to zero. These points represent the solutions or roots of the equation, where the function intersects the x-axis. Zeros are an important concept in various mathematical topics, including the review of functions and Newton's method for finding roots of equations.
Zeros of a function: Zeros of a function are the points where the function's value is equal to zero. These are the \(x\)-values for which \(f(x) = 0\).
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