Finding The Range Of A Function Using A Table A Step-by-Step Guide

Hey guys! Let's dive into how to find the range of a function when we're given a specific domain. We're going to tackle this by using a table, which makes the whole process super organized and easy to follow. So, let's get started!

Understanding the Problem

First, let's break down what we're actually trying to do. We have a function, $f(x) = x^2 - 5x + 6$, and a domain, which is the set of x-values we're allowed to plug into the function. In this case, our domain is $x = \{-2, -1, 0, 1\}$. The range, on the other hand, is the set of all possible y-values (or $f(x)$ values) that we get when we plug in those x-values. So, our mission is to find the range for this particular function and domain. The range of a function is the set of all possible output values (y-values) that the function can produce for the given input values (x-values) within its domain. Essentially, it's the collection of all the results you get when you plug in the allowed x-values into the function's equation. To determine the range, you need to evaluate the function at each x-value in the domain and gather the resulting y-values. These y-values together form the range of the function for that specific domain. In mathematical terms, if you have a function f(x) and a domain X, the range is the set of all f(x) values for every x in X. Think of the domain as the inputs and the range as the outputs of a machine (the function). You put something in (x), the machine processes it, and something comes out (y or f(x)).

Setting Up the Table

To keep things nice and tidy, we'll set up a table. This will help us organize our calculations and make sure we don't miss anything. Our table will have two columns: one for the x-values (from our domain) and one for the corresponding y-values (which we'll calculate using our function). Setting up a table is a smart move when you're trying to find the range of a function for a given domain, and it's a great way to keep your work organized and avoid mistakes. The basic idea is to create a visual structure that maps each x-value from the domain to its corresponding y-value after plugging it into the function. First, you'll want to create columns for your x-values and y-values. List the x-values from your domain in the first column. These are the inputs you're going to use. Next, for each x-value, you'll plug it into the function's equation and calculate the resulting y-value. Write these y-values in the second column, aligned with their corresponding x-values. This process ensures you have a clear record of each input and its output. Tables are especially handy for functions with a small, discrete domain because you can easily see the full set of inputs and their outputs. Plus, having everything laid out in a table makes it easier to spot patterns or any potential errors in your calculations. If you're dealing with more complex functions or larger domains, a table might become cumbersome, but for simple cases like this one, it's an excellent tool for organization and accuracy.

Here’s what our table will look like:

Calculating the y-values

Now comes the fun part – plugging in our x-values and calculating the corresponding y-values. We'll take each x-value from our domain and substitute it into the function $f(x) = x^2 - 5x + 6$. Let's go through each one step-by-step. When calculating the y-values for a function, the process involves substituting each x-value from the domain into the function's equation and then simplifying to find the corresponding y-value. This is a fundamental step in determining the range of the function for a given domain. First, grab an x-value from your domain. For example, if your function is $f(x) = x^2 - 5x + 6$ and your domain includes $x = -2$, you start by replacing every instance of 'x' in the equation with '-2'. This gives you $f(-2) = (-2)^2 - 5(-2) + 6$. Next, simplify the equation using the order of operations (PEMDAS/BODMAS). In this case, you would first handle the exponent, then multiplication, and finally addition and subtraction. So, $(-2)^2$ becomes 4, $-5(-2)$ becomes 10, and the equation simplifies to $4 + 10 + 6$. Finally, add the numbers together to get the y-value. In our example, $4 + 10 + 6 = 20$, so when $x = -2$, $y = 20$. Repeat this process for each x-value in your domain, making sure to take your time and double-check your calculations. It’s easy to make a small arithmetic error, which can throw off your final answer. Once you’ve calculated all the y-values, you'll have the set of outputs that form the range of the function for the given domain.

For x = -2:

$f(-2) = (-2)^2 - 5(-2) + 6 = 4 + 10 + 6 = 20$

For x = -1:

$f(-1) = (-1)^2 - 5(-1) + 6 = 1 + 5 + 6 = 12$

For x = 0:

$f(0) = (0)^2 - 5(0) + 6 = 0 - 0 + 6 = 6$

For x = 1:

$f(1) = (1)^2 - 5(1) + 6 = 1 - 5 + 6 = 2$

Filling in the Table

Now that we've calculated all our y-values, let's fill them into our table:

Identifying the Range

The range is simply the set of all the y-values we calculated. Looking at our table, we can see that the y-values are 20, 12, 6, and 2. So, the range of the function for the given domain is $y = \{20, 12, 6, 2\}$. Identifying the range from a set of calculated y-values is the final step in understanding how a function behaves over a specific domain. Once you've calculated the y-values corresponding to each x-value in the domain, the range is simply the collection of these y-values. However, there are a few key considerations to keep in mind to ensure you accurately identify the range. First, list all the y-values you've calculated. For example, if your calculations yielded y-values of 20, 12, 6, and 2, write them down. Next, the range is typically represented as a set, so you'll want to enclose your y-values in curly braces. In our case, this would look like {20, 12, 6, 2}. The order in which you list the y-values doesn't technically matter, but it's common practice to list them in ascending or descending order for clarity. Additionally, if any y-values are repeated, you only need to include them once in the set. For instance, if you calculated the same y-value twice, don't list it twice in the range. The range represents the unique set of output values. Finally, make sure to double-check your calculations to ensure you haven't made any errors. A mistake in your calculations will obviously lead to an incorrect range. Once you've verified your y-values and represented them correctly as a set, you've successfully identified the range of the function for the given domain.

Choosing the Correct Answer

Now, let's look at the options given in the problem:

A. $y = \{20, 12, 6, 2\}$ B. $y = \{4, 5, 6, 7\}$ C. $y = \{2, 1, 0, -1\}$ D. $y = \{6, 2, 0, 0\}$

Our calculated range matches option A, so that's our answer! Choosing the correct answer from a set of options requires careful comparison between your calculated results and the provided choices. This step is crucial to ensure that you select the accurate solution and avoid any errors. First, take your calculated range, which is the set of y-values you found by plugging the domain values into the function. Make sure you have this set clearly written down and that you've double-checked your calculations to ensure accuracy. Next, review the options provided in the question. Each option will typically present a set of y-values, and your task is to find the one that exactly matches your calculated range. Then, compare your calculated range with each option one by one. Look for the option where all the y-values are identical to the values in your range. The order of the numbers might be different, but the set of values must be the same. For example, if your calculated range is {20, 12, 6, 2}, you would look for an option that contains these exact numbers. Finally, if you find an option that perfectly matches your calculated range, that's the correct answer. If you don't find a perfect match, double-check your calculations and the options to make sure you haven't overlooked anything. Sometimes, a small error can lead to an incorrect selection, so it's always wise to verify your work.

Final Answer

The correct answer is A. $y = \{20, 12, 6, 2\}$.

I hope this helped you guys understand how to find the range of a function using a table! It's all about staying organized and taking it one step at a time. Keep practicing, and you'll master it in no time!