Functions and Graphs (Chapter 1 Summary)
Chapter 1: Functions and Graphs
1.1 Functions (Continued)
Graphing Equations in Two Variables
Point-by-point plotting is a method used to sketch the graph of an equation. This involves plotting a sufficient number of points from its solution set in a rectangular coordinate system until the overall shape of the graph becomes apparent, then connecting these points with a smooth curve.
Definition of a Function
A function is defined as a correspondence between two sets of elements.
Crucially, for each element in the first set (called the domain), there corresponds one and only one element in the second set (called the range).
Variables in Functions
If x acts as a placeholder for elements within the domain of a function, it is designated as the independent variable or input.
If y acts as a placeholder for elements within the range of a function, it is designated as the dependent variable or output.
Determining if an Equation Specifies a Function
An equation in two variables specifies a function if, for every input, exactly one output is produced.
The graph of such a function is identical to the graph of its specifying equation.
If a given input yields more than one output, the equation does not specify a function.
Vertical-Line Test
This test is a graphical method to ascertain whether an equation in two variables specifies a function (as outlined in Theorem 1, page 7 of the textbook).
Types of Basic Functions
Linear Functions: These are functions specified by equations of the form y = mx + b, where m
eq 0.Constant Functions: These are functions specified by equations of the form y = b.
Assumed Domain
When a function is defined by an equation and its domain is not explicitly stated, it is conventionally assumed that the domain consists of all real number inputs that yield real number outputs.
Function Notation
The symbol f(x) denotes the specific element in the range of the function f that corresponds to the given element x from its domain.
Business Applications: Break-Even and Profit-Loss Analysis
This analysis utilizes a cost function (C) and a revenue function (R) to determine financial performance.
A company experiences a loss when R < C.
A company breaks even when R = C.
A company makes a profit when R > C.
Typical forms for cost, revenue, profit, and price-demand functions are detailed on page 11.
Relevant Examples: Ex. 1, p. 3; Ex. 2, p. 6; Ex. 3, p. 8; Ex. 5, p. 9; Ex. 4, p. 9; Ex. 6, p. 10; Ex. 7, p. 11.
1.2 Elementary Functions: Graphs and Transformations
Six Basic Elementary Functions
The graphs of these foundational functions—the identity function, the square function, the cube function, the square root function, the cube root function, and the absolute value function—are presented on page 18.
Graph Transformations
Applying an operation to a function results in a transformation of its graph.
Key basic graph transformations include:
Vertical translations (shifts)
Horizontal translations (shifts)
Reflection in the x-axis
Vertical stretches and shrinks
These transformations are summarized comprehensively on page 22.
Piecewise-Defined Function
This type of function is characterized by having its definition specified by different rules or formulas for distinct parts of its domain.
Relevant Examples: Ex. 1, p. 17; Ex. 2, p. 19; Ex. 3, p. 20; Ex. 4, p. 21; Ex. 5, p. 23; Ex. 6, p. 23; Ex. 7, p. 24.
1.3 Linear and Quadratic Functions
Mathematical Model
A mathematical model is formulated as a mathematics problem designed to be solved, with its solution providing insights and information relevant to a real-world problem.
Linear Equation in Two Variables
This is an equation that can be expressed in the standard form Ax + By = C. In this form, A, B, and C are constants (with the condition that A and B are not both zero), and x and y represent the variables.
The graph of any linear equation in two variables is a straight line.
Conversely, every straight line within a Cartesian coordinate system can be represented by an equation of the form Ax + By = C.
Slope of a Line
If (x1, y1) and (x2, y2) are two distinct points on a line, and x1 eq x2, the slope (m) of the line is calculated as: m = rac{y2 - y1}{x2 - x1}.
Forms of a Line's Equation
Point-slope form: Given a line with slope m that passes through a specific point (x1, y1), its equation can be written as y - y1 = m(x - x1).
Slope-intercept form: A line with slope m and a y-intercept of b has the equation y = mx + b.
Special Line Types
The graph of the equation x = a represents a vertical line.
The graph of the equation y = b represents a horizontal line.
Linear Function
A function of the form f(x) = mx + b, where m
eq 0, is classified as a linear function.
Quadratic Function
A function expressed in the standard form f(x) = ax^2 + bx + c, where a
eq 0, is a quadratic function.The graph of a quadratic function is always a parabola.
By applying the technique of completing the square to the standard form of a quadratic function, it can be converted into the vertex form: f(x) = a(x - h)^2 + k.
Analysis from Vertex Form
The vertex form of a quadratic function (f(x) = a(x - h)^2 + k) directly provides key information, including the coordinates of the vertex, the equation of the axis of symmetry, whether the function has a maximum or minimum value, and the function's range. This form also greatly facilitates sketching the graph (details on pages 37 and 38).
Equilibrium in a Competitive Market
In economic analysis of a competitive market, the intersection point of the supply equation and the demand equation is termed the equilibrium point.
The price corresponding to this point is the equilibrium price.
The common value of supply and demand at this point is the equilibrium quantity.
Regression Analysis for Data Fitting
A scatter plot is a visual representation of the points within a data set.
Linear regression is a statistical method used to find the linear function (a straight line) that provides the best fit for a given data set.
Quadratic regression is a similar method used to determine the quadratic function (a parabola) that best fits a data set.
Relevant Examples: Ex. 1, p. 30; Ex. 2, p. 31; Ex. 3, p. 33; Ex. 4, p. 38; Ex. 6, p. 41; Ex. 5, p. 39; Ex. 7, p. 43; Ex. 8, p. 43.
1.4 Polynomial and Rational Functions
Polynomial Function
A polynomial function can be written in the general form: f(x) = an x^n + a{n-1} x^{n-1} + … + a1 x + a0.
Here, n is a non-negative integer, which defines the degree of the polynomial.
The coefficients a0, a1, …, an are real numbers, with the leading coefficient an
eq 0.The domain of any polynomial function is the set of all real numbers.
Representative graphs of polynomial functions are illustrated on page 51 and in the endpapers of the book.
The graph of a polynomial function of degree n can intersect the x-axis at most n times. These x-intercepts are also referred to as zeros or roots of the function.
A key characteristic of polynomial function graphs is that they possess no sharp corners and are continuous, meaning they have no holes or breaks.
Polynomial regression is a technique that generates a polynomial of a specified degree that optimally fits a given data set.
Rational Function
A rational function is any function that can be expressed in the form: f(x) = rac{n(x)}{d(x)}, where n(x) and d(x) are both polynomial functions, and the denominator d(x) is not equal to zero.
The domain of a rational function encompasses all real numbers, with the exclusion of any values of x for which d(x) = 0.
Representative graphs of rational functions are shown on page 54 and in the endpapers of the book.
Unlike polynomial functions, rational functions can exhibit vertical asymptotes (though not exceeding the degree of the denominator d(x)) and at most one horizontal asymptote.
A detailed procedure for identifying both vertical and horizontal asymptotes of a rational function is provided on page 56.
Relevant Examples: Ex. 1, p. 52; Ex. 2, p. 54; Ex. 3, p. 57; Ex. 4, p. 57.
1.5 Exponential Functions
Definition of an Exponential Function
An exponential function is defined by the form: f(x) = b^x.
Here, b is a positive constant and b
eq 1, referred to as the base.The domain of an exponential function f is the set of all real numbers.
The range of an exponential function is the set of all positive real numbers.
Graph Characteristics
The graph of an exponential function is continuous.
It always passes through the point (0, 1).
The x-axis serves as a horizontal asymptote for the graph.
If the base b > 1, the function b^x is an increasing function as x increases.
If the base 0 < b < 1, the function b^x is a decreasing function as x increases (as stated in Theorem 1, page 63).
Properties of Exponential Functions
Exponential functions adhere to the familiar laws of exponents and satisfy additional specific properties (as detailed in Theorem 2, page 64).
The Natural Base (e)
The irrational number e (approximately 2.7183) is the base most frequently encountered and utilized in mathematical contexts.
Applications of Exponential Functions
Exponential functions are effectively used to model population growth phenomena.
They are also crucial in describing radioactive decay processes.
Exponential regression on a graphing calculator allows for finding the function of the form y = ab^x that best fits a given data set.
Exponential functions are fundamental in calculations involving compound interest and continuous compound interest. The relevant formulas are:
Compound interest: A = Pig(1 + rac{r}{m}ig)^{mt}.
Continuous compound interest: A = Pe^{rt}. (A summary of these concepts is available on page 69).
Relevant Examples: Ex. 1, p. 63; Ex. 2, p. 66; Ex. 3, p. 66; Ex. 4, p. 67; Ex. 5, p. 66; Ex. 6, p. 66.
1.6 Logarithmic Functions
One-to-One Function
A function is classified as one-to-one if each distinct value in its range corresponds to exactly one distinct value in its domain.
Inverse of a Function
The inverse of a one-to-one function f is another function derived by interchanging the roles of the independent and dependent variables of f.
Specifically, if (a, b) is a point on the graph of f, then (b, a) will be a point on the graph of its inverse function.
It is important to note that only functions that are one-to-one possess an inverse function.
Definition of a Logarithmic Function
The inverse of an exponential function with base b is known as the logarithmic function with base b, denoted as y = ext{log}_b x.
The domain of ext{log}_b x is the set of all positive real numbers (which is equivalent to the range of b^x).
The range of ext{log}_b x is the set of all real numbers (which is equivalent to the domain of b^x).
Equivalence between Logarithmic and Exponential Forms
The logarithmic form y = ext{log}_b x is mathematically equivalent to the exponential form x = b^y.
Properties of Logarithmic Functions
The various properties of logarithmic functions are directly derived from and correspond to the properties of exponential functions (as stated in Theorem 1, page 76).
Types of Logarithms
Common logarithms are logarithms with base 10, frequently denoted simply as ext{log } x.
Natural logarithms are logarithms with base e (the irrational number used in exponential functions), and they are frequently denoted as ext{ln } x.
Applications of Logarithmic Functions
Logarithms are used to calculate an investment's doubling time, which is the duration required for the initial value of an investment to double.
Logarithmic regression on a graphing calculator can produce a function of the form y = a + b ext{ ln } x that represents the best fit for a given data set.
Relevant Examples: Ex. 1, p. 74; Ex. 2, p. 75; Ex. 3, p. 75; Ex. 4, p. 76; Ex. 5, p. 76; Ex. 6, p. 77; Ex. 7, p. 77; Ex. 8, p. 78; Ex. 9, p. 78; Ex. 10, p. 79; Ex. 11, p. 80.
Chapter 1: Functions and Graphs
1.1 Functions (Continued)
Graphing Equations in Two Variables
Point-by-point plotting is a way to draw the picture (graph) of an equation. You find several points that work for the equation and mark them on a coordinate system. Once you have enough points, you connect them with a smooth line or curve to see the shape of the graph.
Example: To graph y = x^2, you might plot points like (-2, 4), (-1, 1), (0, 0), (1, 1), and (2, 4), then connect them to form a parabola.
Definition of a Function
A function is a special rule that links two sets of numbers or elements.
For every item in the first set (called the domain), there is only one corresponding item in the second set (called the range).
Example: If a machine takes a number and squares it, for every input like x=3, there's only one output (y=9). This is a function. If for x=3 it could output both 9 and -9, it wouldn't be a function.
Variables in Functions
When we use x to represent values from the domain, x is called the independent variable or input.
When we use y to represent values from the range, y is called the dependent variable or output. The output y depends on the input x.
Determining if an Equation Specifies a Function
An equation with two variables forms a function if each input (x value) always leads to exactly one output (y value).
If an input can give you more than one output, it's not a function.
Example: The equation y = x^2 is a function because for any x, there's only one y. But x^2 + y^2 = 1 (a circle) is NOT a function because for x=0, y could be 1 or -1.
Vertical-Line Test
This is a quick way to check if a graph represents a function: if any vertical line passes through the graph more than once, then the graph does not represent a function.
Types of Basic Functions
Linear Functions: Equations like y = mx + b, where m is not zero. Their graphs are straight lines with a slant.
Example: y = 2x + 3 (a line with slope 2 and y-intercept 3)
Constant Functions: Equations like y = b. Their graphs are flat, horizontal lines.
Example: y = 5 (a horizontal line passing through y=5)
Assumed Domain
If a function's domain isn't explicitly stated, we assume it includes all real numbers that will produce a real number as an output.
Example: For f(x) = \frac{1}{x}, the domain cannot include x=0 because division by zero is undefined. For g(x) = \sqrt{x}, the domain cannot include negative numbers because you can't get a real number output for the square root of a negative.
Function Notation
We use f(x) (read as "f of x") to show the output of the function f when x is the input.
Example: If f(x) = x^2, then f(3) means we put 3 into the function, so f(3) = 3^2 = 9.
Business Applications: Break-Even and Profit-Loss Analysis
This involves a cost function (C) (how much it costs to make something) and a revenue function (R) (how much money you make from selling something).
You have a loss when R < C (you spend more than you earn).
You break even when R = C (you earn exactly what you spend).
You make a profit when R > C (you earn more than you spend).
1.2 Elementary Functions: Graphs and Transformations
Six Basic Elementary Functions
These are fundamental functions whose graphs are building blocks for many other functions (identity, square, cube, square root, cube root, absolute value functions). Their graphs are shown on page 18.
Graph Transformations
When you change a function's rule, its graph changes in specific ways. These changes are called transformations:
Vertical translations (shifts): Moves the graph up or down.
Example: Adding a constant c to f(x) (e.g., f(x) = x^2 + 2 shifts x^2 up by 2 units).
Horizontal translations (shifts): Moves the graph left or right.
Example: Subtracting a constant c inside f(x) (e.g., f(x) = (x - 3)^2 shifts x^2 right by 3 units).
Reflection in the x-axis: Flips the graph upside down.
Example: f(x) = -x^2 flips the parabola y=x^2 across the x-axis.
Vertical stretches and shrinks: Makes the graph taller or flatter.
Example: Multiplying f(x) by a constant a (e.g., f(x) = 2x^2 stretches x^2 vertically; f(x) = \frac{1}{2}x^2 shrinks it).
Piecewise-Defined Function
This is a function that uses different rules (formulas) for different parts of its domain.
Example: The absolute value function f(x) = |x| can be defined as f(x) = x if x \ge 0 and f(x) = -x if x < 0. You use one rule for positive x and another for negative x.
1.3 Linear and Quadratic Functions
Mathematical Model
This is a math problem created to represent and help solve a real-world problem. The solution from the math problem gives us insights into the real-world situation.
Example: Using a linear equation to predict a company's sales based on advertising spending.
Linear Equation in Two Variables
An equation like Ax + By = C. Here, A, B, and C are just numbers (and A and B aren't both zero), and x and y are the variables.
The graph of this type of equation is always a straight line.
Example: 2x + 3y = 6 is a linear equation. Its graph is a straight line.
Slope of a Line
The slope (m) describes how steep a line is. If you have two points (x1, y1) and (x2, y2) on the line, the slope is calculated as: m = \frac{y2 - y1}{x2 - x1} (the change in y divided by the change in x).
Example: For points (1,2) and (3,5), the slope m = \frac{5 - 2}{3 - 1} = \frac{3}{2}.
Forms of a Line's Equation
Point-slope form: If you know the slope (m) and one point (x1, y1) on the line, the equation is y - y1 = m(x - x1).
Example: A line with slope 3/2 passing through (1,2) is y - 2 = \frac{3}{2}(x - 1).
Slope-intercept form: If you know the slope (m) and where the line crosses the y-axis (the y-intercept, b), the equation is y = mx + b.
Example: A line with slope 3 and y-intercept 5 is y = 3x + 5.
Special Line Types
The equation x = a always graphs as a vertical line (e.g., x=2 is a vertical line).
The equation y = b always graphs as a horizontal line (e.g., y=4 is a horizontal line).
Linear Function
A function in the form f(x) = mx + b, where m is not zero, is called a linear function.
Example: f(x) = -4x + 7.
Quadratic Function
A function in the standard form f(x) = ax^2 + bx + c, where a is not zero. Its graph is always a U-shaped curve called a parabola.
Example: f(x) = 2x^2 - 4x + 1.
You can change the standard form into vertex form (f(x) = a(x - h)^2 + k) by using a method called completing the square.
Example: f(x) = 2x^2 - 4x + 1 can be rewritten as f(x) = 2(x - 1)^2 - 1.
Analysis from Vertex Form
The vertex form (f(x) = a(x - h)^2 + k) immediately tells you the vertex (the turning point of the parabola, (h, k)), the axis of symmetry (a line that cuts the parabola in half, x=h), and whether the parabola opens up or down (determining a maximum or minimum value).
Equilibrium in a Competitive Market
This is the point where the supply of a product exactly meets the demand for it. The equilibrium price is the price at which this happens, and the equilibrium quantity is how much is supplied and demanded at that price.
Regression Analysis for Data Fitting
A scatter plot shows all your data points visually.
Linear regression finds the straight line that best fits a set of data points.
Quadratic regression finds the parabola (a U-shaped curve) that best fits a set of data points.
Example: You might use linear regression to find a trend line for a stock's price over time or quadratic regression to model the path of a thrown ball.
1.4 Polynomial and Rational Functions
Polynomial Function
A function written as f(x) = an x^n + a{n-1} x^{n-1} + \dots + a1 x + a0.
n is a whole number (0, 1, 2, \dots) and is the degree (highest power of x).
a0, a1, \dots, an are just numbers, and an (the leading coefficient) is not zero.
The domain (all possible inputs) for any polynomial function is all real numbers.
Its graph is always continuous (no breaks or holes) and smooth (no sharp corners).
The graph can cross the x-axis at most n times (these crossing points are called zeros or roots).
Example: f(x) = 3x^4 - 2x^2 + x - 5 is a polynomial of degree 4.
Polynomial regression is a method that finds a polynomial curve of a specific degree that best fits a given set of data points.
Rational Function
A function that is a fraction, like f(x) = \frac{n(x)}{d(x)}, where both n(x) and d(x) are polynomial functions, and the bottom part (d(x)) is not zero.
The domain includes all real numbers except for any x values that make the denominator (d(x)) equal to zero.
Example: For f(x) = \frac{x+1}{x-2}, the domain is all real numbers except x=2, because if x=2, the denominator would be zero.
Rational functions can have vertical asymptotes (imaginary vertical lines that the graph gets closer and closer to but never touches) and at most one horizontal asymptote (an imaginary horizontal line the graph approaches as x goes to very large or very small numbers).
Example: For f(x) = \frac{1}{x-2}, there is a vertical asymptote at x=2. As x gets closer to 2, the graph shoots up or down.
1.5 Exponential Functions
Definition of an Exponential Function
Defined as f(x) = b^x.
b is a positive number and not equal to 1 (this is called the base).
Example: f(x) = 2^x (base is 2) or f(x) = (1/2)^x (base is 1/2).
The domain (inputs) is all real numbers.
The range (outputs) is all positive real numbers (the output b^x is always positive).
Graph Characteristics
The graph is unbroken (continuous).
It always goes through the point (0, 1) (because b^0 = 1 for any base b).
The x-axis (y=0) is a horizontal asymptote (the graph gets very close to it but never touches or crosses it).
If the base b > 1 (e.g., 2^x), the function increases as x increases (it grows).
If the base 0 < b < 1 (e.g., (1/2)^x), the function decreases as x increases (it shrinks).
Properties of Exponential Functions
They follow the regular rules of exponents (like b^x \times b^y = b^{x+y}).
The Natural Base (e)
An important irrational number, approximately 2.7183, often used as a base in calculus and real-world models. It's called e.
Applications of Exponential Functions
Used to model situations that involve rapid growth or decay:
Population growth: Like how bacteria multiply.
Radioactive decay: How quickly radioactive materials break down (e.g., carbon dating).
Compound interest: How money grows over time with interest added periodically.
Compound interest formula: A = P \left(1 + \frac{r}{m} \right)^{mt}
Continuous compound interest: Money growing constantly.
Continuous compound interest formula: A = Pe^{rt} (where P is the principal, r is the annual interest rate, t is time in years, m is number of times compounded per year, and A is the final amount).
Exponential regression uses a calculator to find the best-fitting exponential function (y = ab^x) for a set of data.
1.6 Logarithmic Functions
One-to-One Function
A function where each unique output value corresponds to exactly one unique input value.
Example: f(x) = x^3 is one-to-one because different inputs always give different outputs. f(x) = x^2 is not one-to-one because both f(2)=4 and f(-2)=4 (two inputs give the same output).
Inverse of a Function
For a one-to-one function, its inverse function is found by switching the roles of its input and output variables. If (a, b) is a point on the graph of function f, then (b, a) will be a point on the graph of its inverse function. Only one-to-one functions have an inverse function.
Definition of a Logarithmic Function
The inverse of an exponential function with base b is called the logarithmic function with base b, written as y = \log_b x.
Example: Since f(x) = 2^x is an exponential function, its inverse is the logarithmic function g(x) = \log_2 x.
The domain of \log_b x is all positive real numbers (this is because it's the range of b^x).
The range of \log_b x is all real numbers (this is because it's the domain of b^x).
Equivalence between Logarithmic and Exponential Forms
The logarithmic form y = \log_b x means the same thing as the exponential form x = b^y.
Example: \log_2 8 = 3 is equivalent to 2^3 = 8.
Properties of Logarithmic Functions
Logarithmic functions follow specific rules that come directly from the rules of exponential functions.
Types of Logarithms
Common logarithms: These use base 10, and are often written simply as \log x.
Natural logarithms: These use the special base e (the natural base, approximately 2.7183), and are written as \ln x.
Applications of Logarithmic Functions
Logarithms are very useful for calculating an investment's doubling time (how long it takes for money to double).
Example: If you want to know how long it takes for an investment to double at a certain interest rate, you would use a logarithm.
Logarithmic regression (using a graphing calculator) can find the best-fitting function of the form y = a + b \ln x for a given data set.