Excluded Values: Find Undefined Points In (t^2 + 10t + 16) / (t^2 - 36)
Hey guys! Today, we're diving into a common topic in algebra: finding excluded values in rational expressions. Specifically, we'll tackle the expression $\frac{t^2 + 10t + 16}{t^2 - 36}$ and figure out for what values of t this expression becomes undefined. This is a crucial skill in algebra because it helps us understand the domain of rational functions, ensuring we don't divide by zero! So, let's break it down step by step.
Understanding Excluded Values
Before we jump into the specifics, let's quickly recap what excluded values actually are. In any rational expression (a fraction where the numerator and denominator are polynomials), we need to be mindful of the denominator. Remember, division by zero is a big no-no in mathematics – it's undefined! So, any value of the variable that makes the denominator equal to zero is an excluded value. These values are excluded from the domain of the function because they would result in an undefined expression.
Think of it like this: if you have a pizza and want to share it with some friends, you can divide it into 2 slices, 4 slices, or even 10 slices. But you can't divide it into 0 slices – that just doesn't make sense! Similarly, in math, we can perform division by any number except zero. That's why identifying and excluding these values is so important.
Now, why is this concept so important? Well, finding excluded values is fundamental in various areas of mathematics. It helps us define the domain of rational functions, solve rational equations, and understand the behavior of graphs. When we're graphing rational functions, the excluded values often correspond to vertical asymptotes, which are lines that the graph approaches but never actually touches. So, mastering this concept opens the door to a deeper understanding of algebraic functions and their applications.
Step-by-Step Guide to Finding Excluded Values
Okay, let's get practical and find the excluded values for our given expression: $\frac{t^2 + 10t + 16}{t^2 - 36}$. Here’s the breakdown:
1. Focus on the Denominator
The golden rule for finding excluded values is: ignore the numerator and zero in on the denominator. The numerator can be anything – it doesn't affect whether the expression is defined or not. The denominator, however, is the key. In our case, the denominator is $t^2 - 36$. We need to find the values of t that make this expression equal to zero.
2. Set the Denominator Equal to Zero
This is the most crucial step. We take our denominator and set it equal to zero:$t^2 - 36 = 0$ This equation represents the condition we need to solve to find our excluded values. We're essentially asking: "For what values of t does this equation hold true?"
3. Solve for the Variable
Now we have a simple quadratic equation to solve. There are a couple of ways we can tackle this. One common method is factoring. Another method is using the square root property. Let's use the factoring method first.
Factoring Method
Notice that $t^2 - 36$ is a difference of squares. We can factor it as: $(t - 6)(t + 6) = 0$ This factorization is based on the algebraic identity $a^2 - b^2 = (a - b)(a + b)$. In our case, $a = t$ and $b = 6$. Now, we have a product of two factors equal to zero. This means that at least one of the factors must be zero. So, we set each factor equal to zero and solve:
Square Root Property Method
Alternatively, we could have used the square root property. Starting from $t^2 - 36 = 0$, we can add 36 to both sides:$t^2 = 36$ Now, we take the square root of both sides. Remember, when we take the square root, we need to consider both the positive and negative roots:$\sqrt{t^2} = \pm\sqrt{36}$$t = \pm 6$ This gives us the same two solutions: t = 6 and t = -6.
4. Identify the Excluded Values
We've done the math, and we've found the values of t that make the denominator zero. These are our excluded values! So, the excluded values for the expression $\frac{t^2 + 10t + 16}{t^2 - 36}$ are t = 6 and t = -6. These are the values that we need to exclude from the domain of the function.
Let's Summarize with an Example
To make sure we've got this down, let's recap with our example expression. We started with $\frac{t^2 + 10t + 16}{t^2 - 36}$. We identified the denominator as $t^2 - 36$. We set the denominator equal to zero: $t^2 - 36 = 0$. We then solved for t using either factoring or the square root property, and we found that $t = 6$ and $t = -6$ are the values that make the denominator zero. Therefore, 6 and -6 are the excluded values for this expression.
Why This Matters: Connecting to the Big Picture
Now that we know how to find excluded values, let's zoom out and see why this skill is so important in the grand scheme of mathematics. As we touched on earlier, excluded values are intrinsically linked to the concept of the domain of a function. The domain is essentially the set of all possible input values (in our case, t values) for which the function is defined. Excluded values are the troublemakers – the inputs that cause the function to misbehave (by resulting in division by zero). By identifying these values, we can accurately describe the function's domain.
For the expression $\fract^2 + 10t + 16}{t^2 - 36}$, the domain would be all real numbers except 6 and -6. We can express this mathematically using set notation or interval notation. In set notation, we might write \mid t \neq 6, t \neq -6}$ This reads as "the set of all t in the real numbers such that t is not equal to 6 and t is not equal to -6." In interval notation, we would write:$(-\infty, -6) \cup (-6, 6) \cup (6, \infty)$ This notation indicates three intervals: all numbers less than -6, all numbers between -6 and 6, and all numbers greater than 6. The union symbol ($\cup$) combines these intervals to represent the entire domain.
Furthermore, excluded values play a crucial role in understanding the graphs of rational functions. At excluded values, we often find vertical asymptotes. A vertical asymptote is a vertical line that the graph of the function approaches but never intersects. These asymptotes act as boundaries, shaping the overall behavior of the graph. When we graph $\frac{t^2 + 10t + 16}{t^2 - 36}$, we'd see vertical asymptotes at $t = 6$ and $t = -6$. These asymptotes tell us that the function's output becomes infinitely large (either positive or negative) as t gets closer to these excluded values.
Practice Makes Perfect: Try These Examples
Okay, guys, now it's your turn to practice! Here are a few more rational expressions. Try finding the excluded values for each:
Remember the key steps: focus on the denominator, set it equal to zero, solve for the variable, and identify those values as your excluded values. Don't be afraid to factor, use the quadratic formula, or any other algebraic technique you have in your toolbox!
Conclusion
Finding excluded values might seem like a small detail, but it's a fundamental concept that underpins a lot of what we do in algebra and beyond. By mastering this skill, we can better understand the domain of rational functions, solve equations, and interpret graphs. It's all about understanding the rules of the game – in this case, the rule that division by zero is a mathematical no-go. Keep practicing, and you'll become a pro at spotting those excluded values in no time! You've got this! Happy calculating!
