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Cp1 Algebra 2 Homework Ws 32: Learn the Basics of Domain and Range with Examples and Exercises


Cp1 Algebra 2 Homework Ws 32: Learn the Basics of Domain and Range with Examples and Exercises




Do you struggle with finding the domain and range of a function? Do you want to learn how to identify the possible input and output values of a function given by an equation, a graph, or a table? If so, this article is for you.




Cp1 Algebra 2 Homework Ws 32 Domain And Range



In this article, you will learn what domain and range are, how to find them using different methods, and how to apply them to real-world situations. You will also find some examples and exercises to practice your skills and check your understanding.


What are Domain and Range?




The domain of a function is the set of all possible input values for the function. The range of a function is the set of all possible output values for the function. In other words, the domain is what you can put into the function, and the range is what you can get out of the function.


For example, consider the function f(x) = x + 2. This function takes any real number x as an input and adds 2 to it. The output is also a real number. Therefore, the domain and range of this function are both all real numbers.


Another way to think about domain and range is to imagine a function as a machine that transforms inputs into outputs. The domain is what you can feed into the machine, and the range is what you can collect from the machine. See Figure 1 for an illustration.


Figure 1: A function machine with domain and range


How to Find the Domain and Range of a Function?




There are different methods to find the domain and range of a function depending on how the function is given. Here are some common methods:


  • If the function is given by an equation, we can use algebraic techniques to find the domain and range. For example, we can look for values that would make the denominator zero or the radicand negative in the equation.



  • If the function is given by a graph, we can use visual techniques to find the domain and range. For example, we can look for the lowest and highest values of x and y on the graph.



  • If the function is given by a table, we can use numerical techniques to find the domain and range. For example, we can look for the smallest and largest values of x and y in the table.



In this article, we will focus on finding the domain and range of a function given by an equation using algebraic techniques.


Finding the Domain of a Function Defined by an Equation




To find the domain of a function defined by an equation, we need to consider what values of x are allowed or excluded by the equation. There are three main cases to consider:


  • If the function has no denominator or an odd root, then the domain is usually all real numbers.



  • If the function has a denominator, then we need to exclude any values of x that would make the denominator zero.



  • If the function has an even root, then we need to exclude any values of x that would make the radicand negative.



Let's look at some examples of finding the domain of a function defined by an equation.


Example 1




Find the domain of the function f(x) = 1/(x - 2).


Solution:


This function has a denominator, so we need to exclude any value of x that would make the denominator zero. To find such value, we set the denominator equal to zero and solve for x:


x - 2 = 0


x = 2


Therefore, x = 2 is not in the domain of the function. The domain of the function is all real numbers except for x = 2. We can write this in interval notation as:


(-, 2) (2, )


Example 2




Find the domain of the function g(x) = (x + 5).


Solution:


This function has an even root, so we need to exclude any value of x that would make the radicand negative. To find such values, we set the radicand greater than or equal to zero and solve for x:


x + 5 0


x -5


Therefore, x -5 is in the domain of the function. The domain of the function is all real numbers that are greater than or equal to -5. We can write this in interval notation as:


[-5, )


Finding the Range of a Function Defined by an Equation




To find the range of a function defined by an equation, we need to consider what values of y are possible or impossible for the function. There are different methods to find the range of a function depending on the type of the function. Here are some common methods:


  • If the function is linear, such as f(x) = 2x + 5, then the range is usually all real numbers.



  • If the function is quadratic, such as f(x) = x^2 - 4, then the range has a minimum or maximum value depending on whether the parabola opens up or down. We can use the vertex form of the quadratic function to find this value.



  • If the function is rational, such as f(x) = 1/(x - 2), then the range may have a horizontal asymptote or a hole depending on whether the numerator and denominator have a common factor. We can use long division or synthetic division to find this value.



  • If the function is radical, such as f(x) = (x + 5), then the range may have a lower or upper bound depending on whether the index of the radical is even or odd. We can use inverse functions to find this value.



Let's look at some examples of finding the range of a function defined by an equation.


Example 3




Find the range of the function f(x) = x^2 - 4.


Solution:


This function is quadratic, so it has a parabola shape. To find the range, we need to find the minimum or maximum value of y for this function. To do this, we can use the vertex form of the quadratic function, which is f(x) = a(x - h)^2 + k, where (h,k) is the vertex and a is a constant that determines whether the parabola opens up or down.


To write f(x) = x^2 - 4 in vertex form, we need to complete the square. Here are the steps:


f(x) = x^2 - 4


f(x) = (x^2 - 0x) - 4


f(x) = (x^2 - 0x + (-0/2)^2) - 4 - (-0/2)^2


f(x) = (x - 0)^2 - 4 - 0


f(x) = (x - 0)^2 - 4


Now we can see that the vertex form of f(x) is f(x) = (x - 0)^2 - 4, where a = 1, h = 0, and k = -4. This means that the vertex of the parabola is (0,-4) and that it opens up because a > 0. Therefore, the minimum value of y for this function is -4 and there is no maximum value. The range of this function is y -4. We can write this in interval notation as:


[-4, ) a27c54c0b2


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