Module 2 topic 1 notes
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The document reviews key concepts about functions including domain, range, and evaluating functions. It provides examples of determining if a relation is a function using mapping diagrams and the vertical line test. It also gives examples of finding the domain and range of functions from graphs and equations. Practice problems are included for students to determine domains, ranges, and evaluate functions.
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![[object Object],Let’s take a minute and do a more thorough review of these concepts. Go now to http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=133 , login, and launch the Gizmo Introduction to Functions. Open the Exploration Guide and follow the instructions.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Lesson 1 functions as models
Hello everyone. I hope this presentation will be helpful for those who are learning General Mathematics Grade 11.
Functions and Relations
This document discusses functions and relations. It defines a relation as a set of ordered pairs that associates each element of one set with an element of another set. A function is a special type of relation where each element of the first set is mapped to exactly one element of the second set. The document provides examples of relations and functions using sets, tables, graphs and equations. It describes key characteristics of functions, such as the vertical line test, and discusses classifying relations as functions or not functions. The learning objectives are for students to illustrate relations and functions, verify if a relation is a function, and determine dependent and independent variables.
PPt on Functions
1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
Composite functions
The document discusses operations and composition of functions. It explains that to find the sum of two functions f and g, you add them together and combine like terms. To find the difference, you subtract the second function from the first and distribute negatives. To find the product, you multiply corresponding terms of f and g. For the quotient f/g, you divide the first function by the second. The domain of sums, differences and products is where x is in the domains of both f and g, while the domain of the quotient excludes values where the denominator would be 0. Composition means substituting one function into another, written as f(g) or g(f).
Evaluating Functions
The document discusses evaluating functions by replacing the variable with a value from the domain and computing the result. It provides examples of evaluating various functions at different values of x. These include evaluating f(x) = 2x + 1 at x = 1.5, q(x) = x^2 - 2x + 2 at x = 2, and other functions at different values. It also discusses cases where a function cannot be evaluated, such as when the value is not in the domain of the function.
Relations and functions
The document provides information about relations and functions:
- It defines key terms like relation, domain, range, ordered pair, and function.
- It explains how to represent relations using ordered pairs, tables, graphs, and mappings.
- It discusses how to determine if a relation is a function using the vertical line test or by checking if each domain value is mapped to only one range value.
- It distinguishes between independent and dependent variables in functional notation.
Lesson 1 INTRODUCTION TO FUNCTIONS
The document defines functions and relations. It reviews basic math operations and introduces the concepts of relations, inputs and outputs, domains and ranges. A relation is a correspondence between two sets, while a function is a special type of relation where each input has a single, unique output. Functions can be illustrated through sets of ordered pairs, mapping diagrams, and graphs where the vertical line test determines if a relation is a function. Students are given examples of relations and functions to identify which are functions.
Graphs of polynomial functions
The document summarizes key characteristics of polynomial functions:
1) Polynomial functions produce smooth, continuous curves on their domains which are the set of real numbers.
2) The graph's x-intercepts, turning points, and absolute/relative maxima and minima are defined.
3) As the degree of a polynomial increases, so do the possible number of x-intercepts and turning points, up to the degree value. The leading coefficient and degree determine whether the graph rises or falls.
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Lesson 1 functions as models
Hello everyone. I hope this presentation will be helpful for those who are learning General Mathematics Grade 11.
Functions and Relations
This document discusses functions and relations. It defines a relation as a set of ordered pairs that associates each element of one set with an element of another set. A function is a special type of relation where each element of the first set is mapped to exactly one element of the second set. The document provides examples of relations and functions using sets, tables, graphs and equations. It describes key characteristics of functions, such as the vertical line test, and discusses classifying relations as functions or not functions. The learning objectives are for students to illustrate relations and functions, verify if a relation is a function, and determine dependent and independent variables.
PPt on Functions
1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
Composite functions
The document discusses operations and composition of functions. It explains that to find the sum of two functions f and g, you add them together and combine like terms. To find the difference, you subtract the second function from the first and distribute negatives. To find the product, you multiply corresponding terms of f and g. For the quotient f/g, you divide the first function by the second. The domain of sums, differences and products is where x is in the domains of both f and g, while the domain of the quotient excludes values where the denominator would be 0. Composition means substituting one function into another, written as f(g) or g(f).
Evaluating Functions
The document discusses evaluating functions by replacing the variable with a value from the domain and computing the result. It provides examples of evaluating various functions at different values of x. These include evaluating f(x) = 2x + 1 at x = 1.5, q(x) = x^2 - 2x + 2 at x = 2, and other functions at different values. It also discusses cases where a function cannot be evaluated, such as when the value is not in the domain of the function.
Relations and functions
The document provides information about relations and functions:
- It defines key terms like relation, domain, range, ordered pair, and function.
- It explains how to represent relations using ordered pairs, tables, graphs, and mappings.
- It discusses how to determine if a relation is a function using the vertical line test or by checking if each domain value is mapped to only one range value.
- It distinguishes between independent and dependent variables in functional notation.
Lesson 1 INTRODUCTION TO FUNCTIONS
The document defines functions and relations. It reviews basic math operations and introduces the concepts of relations, inputs and outputs, domains and ranges. A relation is a correspondence between two sets, while a function is a special type of relation where each input has a single, unique output. Functions can be illustrated through sets of ordered pairs, mapping diagrams, and graphs where the vertical line test determines if a relation is a function. Students are given examples of relations and functions to identify which are functions.
Graphs of polynomial functions
The document summarizes key characteristics of polynomial functions:
1) Polynomial functions produce smooth, continuous curves on their domains which are the set of real numbers.
2) The graph's x-intercepts, turning points, and absolute/relative maxima and minima are defined.
3) As the degree of a polynomial increases, so do the possible number of x-intercepts and turning points, up to the degree value. The leading coefficient and degree determine whether the graph rises or falls.
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This document contains a sample lesson plan on general mathematics for grade 11 that includes content on ordered pairs, functions, and determining whether a relation represents a function. The lesson includes examples of sets of ordered pairs and asks students to identify which are functions. It also contains graphs and asks students to determine if the relation is a function using the vertical line test. Key points covered are the definition of a function as a set of ordered pairs where no two pairs have the same first element and how to identify functions from graphs or sets of ordered pairs.
Inverse functions
The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.
Relations and functions
A relation is a set of ordered pairs that can be represented as a table, mapping, or graph. The domain is the set of first elements (x-coordinates) and the range is the set of second elements (y-coordinates). A function is a relation where each element of the domain is paired with exactly one element of the range, with no repeating x-values. Functions can be identified using the vertical line test on their graph, and can be written in function notation f(x) or evaluated by substitution.
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This document defines key concepts related to functions including domain, co-domain, and range. It provides examples of determining the domain and range of various functions. The domain of a function is the set of inputs, the co-domain is the set of all possible outputs, and the range is the set of actual outputs. Examples show how to use inequalities to find the domain by determining when a function gives real values and how to manipulate equations to find the range.
Piecewise Functions
This document discusses piecewise functions and step functions. Piecewise functions are represented by a combination of equations, with each equation corresponding to a part of the domain. Step functions use separate equations for different intervals of the domain, resulting in a graph that makes distinct jumps at the boundaries between intervals. Examples are provided of evaluating piecewise functions and graphing both piecewise and step functions. Special types of step functions called ceiling functions and floor functions are also introduced, which round non-integer values up or down to the nearest integer.
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This document defines and provides examples of the domain and range of a function. The domain is the set of all independent variable (x-coordinate) values, while the range is the set of all dependent variable (y-coordinate) values. Several examples of identifying the domain and range of functions given their sets of ordered pairs are provided. The document also discusses using the vertical line test to determine if a graph represents a function.
Inverse Functions
1) The document discusses finding the inverse of functions by interchanging the x and y variables and solving for y. It provides examples of finding the inverses of f(x)=3x-7 and g(x)=2x^3+1.
2) It also discusses verifying inverses by checking if the composition of a function and its inverse equals x. And finding inverses of functions with restricted domains, including an example of f(x)=sqrt(x+4).
3) Finally, it discusses the relationship between a function being one-to-one and having an inverse function, both algebraically and graphically.
Relations and functions
The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of first coordinates and the range is the set of second coordinates. A function is a special type of relation where each element in the domain corresponds to exactly one element in the range. Functions can be represented as tables of values, ordered pairs, mappings, graphs or equations. The vertical line test can be used to determine if a graph represents a function - if no vertical line intersects the graph at more than one point, it is a function. Examples are provided to illustrate these concepts.
Solving systems of Linear Equations
There are three possible solutions to a system of linear equations in two variables:
One solution: the graphs intersect at a single point, giving the solution coordinates.
No solution: the graphs are parallel lines, making the system inconsistent.
Infinitely many solutions: the graphs are the same line, making the equations dependent.
The substitution method for solving systems involves: 1) solving one equation for a variable, 2) substituting into the other equation, 3) solving the new equation, and 4) back-substituting to find the remaining variable.
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The document provides an overview of key concepts about functions for senior high school mathematics. It discusses representing real-life situations using functions, including piecewise functions. Different ways of representing functions are presented, such as graphs. Examples of relating scenarios to the independent and dependent variables of a function are given. The document also demonstrates representing real-life situations as mathematical function models and provides sample problems about piecewise functions.
Evaluating functions
The document discusses evaluating functions. It begins by defining the difference between a function and a relation. It then outlines four ways to represent a function: sets of ordered pairs, tables of values, graphs, and equations. It explains that evaluating a function means replacing the variable x in the function with a given number or expression. Examples are provided to demonstrate how to evaluate different types of functions by performing the substitution. The document concludes with a quiz asking the reader to evaluate sample functions at specific values of x.
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The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
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The document discusses linear equations in two variables. It defines a linear equation as one that can be written in the standard form Ax + By = C, where A, B, and C are real numbers and A and B cannot both be zero. Examples are provided of determining if equations are linear and identifying the A, B, and C components if they are linear. The document also discusses using ordered pairs as solutions to linear equations and finding multiple solutions to a given linear equation.
Exponential equations
An exponential equation is one where a variable appears in the exponent. To solve exponential equations without logarithms, both sides must have comparable exponential expressions with the same base, so the powers can be set equal. If both sides can be expressed as powers of the same base b, the property b^x = b^y implies x = y can be used to set the exponents equal and solve for the variable. This demonstrates how exponential equations are solved by setting the powers equal when the bases are the same. For equations that cannot be expressed with a common base, logarithms are used.
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The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
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This document contains a sample lesson plan on general mathematics for grade 11 that includes content on ordered pairs, functions, and determining whether a relation represents a function. The lesson includes examples of sets of ordered pairs and asks students to identify which are functions. It also contains graphs and asks students to determine if the relation is a function using the vertical line test. Key points covered are the definition of a function as a set of ordered pairs where no two pairs have the same first element and how to identify functions from graphs or sets of ordered pairs.
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The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.
Relations and functions
A relation is a set of ordered pairs that can be represented as a table, mapping, or graph. The domain is the set of first elements (x-coordinates) and the range is the set of second elements (y-coordinates). A function is a relation where each element of the domain is paired with exactly one element of the range, with no repeating x-values. Functions can be identified using the vertical line test on their graph, and can be written in function notation f(x) or evaluated by substitution.
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This document defines key concepts related to functions including domain, co-domain, and range. It provides examples of determining the domain and range of various functions. The domain of a function is the set of inputs, the co-domain is the set of all possible outputs, and the range is the set of actual outputs. Examples show how to use inequalities to find the domain by determining when a function gives real values and how to manipulate equations to find the range.
Piecewise Functions
This document discusses piecewise functions and step functions. Piecewise functions are represented by a combination of equations, with each equation corresponding to a part of the domain. Step functions use separate equations for different intervals of the domain, resulting in a graph that makes distinct jumps at the boundaries between intervals. Examples are provided of evaluating piecewise functions and graphing both piecewise and step functions. Special types of step functions called ceiling functions and floor functions are also introduced, which round non-integer values up or down to the nearest integer.
Linear Systems - Domain & Range
This document defines and provides examples of the domain and range of a function. The domain is the set of all independent variable (x-coordinate) values, while the range is the set of all dependent variable (y-coordinate) values. Several examples of identifying the domain and range of functions given their sets of ordered pairs are provided. The document also discusses using the vertical line test to determine if a graph represents a function.
Inverse Functions
1) The document discusses finding the inverse of functions by interchanging the x and y variables and solving for y. It provides examples of finding the inverses of f(x)=3x-7 and g(x)=2x^3+1.
2) It also discusses verifying inverses by checking if the composition of a function and its inverse equals x. And finding inverses of functions with restricted domains, including an example of f(x)=sqrt(x+4).
3) Finally, it discusses the relationship between a function being one-to-one and having an inverse function, both algebraically and graphically.
Relations and functions
The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of first coordinates and the range is the set of second coordinates. A function is a special type of relation where each element in the domain corresponds to exactly one element in the range. Functions can be represented as tables of values, ordered pairs, mappings, graphs or equations. The vertical line test can be used to determine if a graph represents a function - if no vertical line intersects the graph at more than one point, it is a function. Examples are provided to illustrate these concepts.
Solving systems of Linear Equations
There are three possible solutions to a system of linear equations in two variables:
One solution: the graphs intersect at a single point, giving the solution coordinates.
No solution: the graphs are parallel lines, making the system inconsistent.
Infinitely many solutions: the graphs are the same line, making the equations dependent.
The substitution method for solving systems involves: 1) solving one equation for a variable, 2) substituting into the other equation, 3) solving the new equation, and 4) back-substituting to find the remaining variable.
Lesson 1: Functions as Models
The document provides an overview of key concepts about functions for senior high school mathematics. It discusses representing real-life situations using functions, including piecewise functions. Different ways of representing functions are presented, such as graphs. Examples of relating scenarios to the independent and dependent variables of a function are given. The document also demonstrates representing real-life situations as mathematical function models and provides sample problems about piecewise functions.
Evaluating functions
The document discusses evaluating functions. It begins by defining the difference between a function and a relation. It then outlines four ways to represent a function: sets of ordered pairs, tables of values, graphs, and equations. It explains that evaluating a function means replacing the variable x in the function with a given number or expression. Examples are provided to demonstrate how to evaluate different types of functions by performing the substitution. The document concludes with a quiz asking the reader to evaluate sample functions at specific values of x.
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The document provides information about relations and functions. It defines domain as the x-coordinates of ordered pairs in a relation, and range as the y-coordinates. An example relation is given with domain {3, 1, -2} and range {2, 6, 0}. Relations can be represented as sets of ordered pairs, tables, mappings, and graphs. A mapping example is shown. Determining if a relation is a function involves checking if each x-value is paired with a single y-value using the vertical line test.
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The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
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Provides example and solutions about about how to find the roots of a quadratic equation and its sum and product.
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The document discusses linear equations in two variables. It defines a linear equation as one that can be written in the standard form Ax + By = C, where A, B, and C are real numbers and A and B cannot both be zero. Examples are provided of determining if equations are linear and identifying the A, B, and C components if they are linear. The document also discusses using ordered pairs as solutions to linear equations and finding multiple solutions to a given linear equation.
Exponential equations
An exponential equation is one where a variable appears in the exponent. To solve exponential equations without logarithms, both sides must have comparable exponential expressions with the same base, so the powers can be set equal. If both sides can be expressed as powers of the same base b, the property b^x = b^y implies x = y can be used to set the exponents equal and solve for the variable. This demonstrates how exponential equations are solved by setting the powers equal when the bases are the same. For equations that cannot be expressed with a common base, logarithms are used.
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The document defines key concepts relating to functions and relations:
- A relation is a set of ordered pairs where the domain is the set of all x-values and the range is the set of all y-values.
- A function is a special type of relation where each x-value is assigned to exactly one y-value.
- Function notation uses f(x) to represent the output of a function f when the input is x.
- The domain of a function is the set of all valid input values that do not result in undefined outputs like division by zero or square roots of negative numbers.
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This document discusses domain and range for different types of functions including linear, quadratic, rational, and irrational functions. It provides examples of finding the domain and range for various functions. The key points are:
- The domain of a function is the set of valid x-values, while the range is the set of y-values.
- For linear, quadratic, and irrational functions, the domain is typically all real numbers, while the range depends on the specific function.
- For rational functions, the domain excludes values that make the denominator equal to zero, while the range typically includes all real numbers.
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Function powerpoinr.pptx
1. Addition of functions: Add the outputs of two functions with the same inputs.
2. Subtraction of functions: Subtract the outputs of two functions with the same inputs.
3. Multiplication of functions: Multiply the outputs of two functions with the same inputs.
4. Division of functions: Divide the output of one function by the output of another function with the same inputs.
5. Composition of functions: Replace the inputs of one function with the outputs of another function.
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The document provides information about functions and relations. It defines a function as a relation where each x-value is paired with exactly one y-value. To determine if a relation is a function, it describes using the vertical line test, where a relation is a function if a vertical line can only intersect the graph at one point. It gives examples of applying the vertical line test to graphs and determining the domain and range of relations.
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Note introductions of functions
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
Introduction to functions
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
function
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
relationsandfunctionsupdated-140102120840-phpapp01.ppt
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Domain and range (linear, quadratic, rational functions)
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relationsandfunctionsupdated-140102120840-phpapp01.ppt
relationsandfunctionsupdated-140102120840-phpapp01.ppt
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Module 2 topic 1 notes
- 1. Module 2 Topic 1
- 7. Practice Try these problems on your own. Once you have answered a problem, click once to see the answer. Remember to work each problem first, then click to see the answer. 3(7) – 4 21 – 4 17 5 2 + 2(5) + 1 25 + 10 + 1 36 f(x) = 3x – 4 for x = 7 f(x) = x 2 + 2x + 1 for x = 5
- 9. In Algebra 1 you learned the basics of domain and range. Click on the link below to watch a video and refresh your memory on that topic. Pearson Prentice Hall Mathematics Video
- 10. Many functions can be represented by an equation in two variables. The input variable is called the independent variable. The output variable in an equation, which depends on the value of the input variable, is called the dependent variable. The domain of a function consists of the set of all input values. The range of a function consist of the set of all output values. A function is like a machine that you input numbers and variables. The machine alters the input in some way and produces an answer.
- 11. Example: Find the domain and range of the following graph: What's the domain ? The graph above is represented by y = x 2 , and we can square any number we want. Therefore, the domain is all real numbers. On a graph the domain corresponds to the horizontal axis. Since that is the case, we need to look to the left and right to see if there are any end points or holes in the graph to help us find our domain. If the graph keeps going on and on to the right and the graph keeps going on and on to the left then the domain is represented by all real numbers.
- 12. What's the range ? If I plug any number into this function, am I ever going to be able to get a negative number out of it? No, not in the Real Number System! The range of this function is all positive numbers which is represented by y ≥ 0. On a graph, the range corresponds to the vertical axis. Since that is the case, we need to look up and down to see if there are any end points or holes to help us find our range. If the graph keeps going up and down with no endpoint then the range is all real numbers. However, this is not the case. The graph does not begin to touch the y-axis until x = 0, then it continues up with no endpoints which is represented by y ≥ 0.
- 13. Let's try another example: What numbers can we plug into this function? What happens if we plug in 4? If x = 4, we divide by zero which is undefined. Therefore, the domain of this function is: all real numbers except 4. The range is all real numbers except 0 . (We can only produce zero when a zero is in the numerator.) , zero in the numerator is OK! However, , zero in the denominator is a NO NO! In general, when you're trying to find the domain of a function, there are two things you should look out for. (1) Look for potential division by zero. (2) Look for places where you might take the square root of a negative number. (3) If you have a verbal model, you can only use numbers that make sense in the given situation. (This is called the relevant domain as it only identifies the values that make sense in the given situation.)
- 15. Summary The domain of a function is all the possible input values, and the range is all possible output values.