Domain Of A Rational Function: Step-by-Step Solution

Oct 25, 2025 by SLV Team 53 views

Hey guys! Let's break down this math problem together. We've got a function and need to figure out its domain. Basically, the domain is all the possible 'x' values that we can plug into the function without causing any mathematical mayhem, like dividing by zero. In this case, our function is a rational function, which is just a fancy way of saying it's a fraction where the top and bottom are polynomials. Specifically, we're looking at:

y = (x^2 − 7x + 10) / (2x + 7)

Understanding the Problem

The key thing to remember with rational functions is that the denominator (the bottom part of the fraction) cannot be zero. If it is, we're dividing by zero, which is a big no-no in math. It makes the function undefined at that point. So, our mission is to find any 'x' values that would make the denominator zero and exclude them from our domain.

Why is this so important? Think of it this way: the domain is like the guest list for a party, and dividing by zero is like showing up in your pajamas – it's just not allowed! We need to make sure our 'x' values are well-behaved and don't cause any trouble. This ensures our function behaves predictably and gives us real, usable outputs.

The numerator, x^2 − 7x + 10, doesn't cause us any domain issues on its own. Polynomials like this are defined for all real numbers. It's that denominator, 2x + 7, that we need to watch out for. This is where the potential for division by zero lurks, and it's our job to sniff it out and eliminate it from the domain. Therefore, the entire problem hinges on what values of x make the denominator zero.

Finding the Forbidden 'x' Value

To find the 'x' value that makes the denominator zero, we set the denominator equal to zero and solve for 'x':

2x + 7 = 0

Subtract 7 from both sides:

2x = -7

Divide both sides by 2:

x = -7/2

So, x = -7/2 is the value that makes our denominator zero. This is the troublemaker we need to exclude from our domain. It's like that one guest who always spills punch on the rug – we love them, but they can't come to this party (domain!).

Let's think about why this is the only value we need to worry about. The denominator is a linear expression. Linear expressions only equal zero at one point. Therefore, there is only one value of x that will cause this rational expression to be undefined. Any other real number will work just fine. This is super important because it helps us define our domain accurately.

Defining the Domain

Okay, now that we know x = -7/2 is off-limits, we can define the domain. The domain is all real numbers except for -7/2. We can write this in a few different ways.

Set Notation: This is a formal way to say it: {x ∈ ℝ | x ≠ -7/2}. This translates to "the set of all x belonging to the real numbers such that x is not equal to -7/2". It is a concise and precise way to define the domain.
Interval Notation: This is a more visual way to represent the domain: (-∞, -7/2) ∪ (-7/2, ∞). This means all numbers from negative infinity up to (but not including) -7/2, and all numbers from -7/2 (but not including) to positive infinity. The ∪ symbol means "union," which combines these two intervals.
R \ {-7/2}: This notation means all real numbers (R) except for the set containing -7/2. It is a very common and efficient way to represent the domain when you're excluding only a single point.

All three of these notations say the same thing: we can use any real number for 'x' except -7/2. Choose the notation that makes the most sense to you and that your teacher or textbook prefers.

Choosing the Correct Answer

Looking back at the choices, the correct answer is:

(A) 𝑅 \ {−7/2}

Because this notation directly states that the domain is all real numbers except for -7/2.

Why the Other Options Are Wrong

Let's quickly look at why the other options are incorrect:

(B) (-7/2, +[infinity]): This interval only includes numbers greater than -7/2. It excludes all numbers less than or equal to -7/2, which are perfectly valid in our function.
(C) [-7/2, +[infinity]): This interval includes -7/2, which we know makes the denominator zero and makes the function undefined.
(D) (2, 5): This interval only includes numbers between 2 and 5. This is far too restrictive and doesn't represent the entire domain of the function.
(E) ∅: This represents the empty set, meaning there are no possible values for x. This is clearly incorrect since we know many values of x work in the function.

Key Takeaways

Rational functions have denominators that cannot be zero. Always check for values of x that make the denominator zero.
The domain is the set of all possible x-values that make the function defined. Exclude any values that cause division by zero or other mathematical errors.
Use set notation, interval notation, or R \ {value} to express the domain. Choose the notation that is clearest and most appropriate for the context.

Let's Practice!

Okay, guys, to really nail this down, let's try a similar problem:

What is the domain of the function f(x) = (3x + 1) / (x - 4)?

Identify the denominator: The denominator is x - 4.
Set the denominator equal to zero: x - 4 = 0
Solve for x: x = 4
Exclude this value from the domain: The domain is all real numbers except for x = 4.
Write the domain in interval notation: (-∞, 4) ∪ (4, ∞)

See? Once you get the hang of it, these problems become much easier!

A More Complex Example

Let's level up and try one where the denominator is a quadratic:

What is the domain of the function g(x) = 5 / (x^2 - 9)?

Identify the denominator: The denominator is x^2 - 9.
Set the denominator equal to zero: x^2 - 9 = 0
Solve for x: This is a difference of squares, so it factors as (x - 3)(x + 3) = 0. Therefore, x = 3 or x = -3.
Exclude these values from the domain: The domain is all real numbers except for x = 3 and x = -3.
Write the domain in interval notation: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)

In this case, we had two values to exclude from the domain because the quadratic denominator had two roots. This highlights the importance of carefully solving for the roots of the denominator.

Conclusion

Finding the domain of a rational function might seem tricky at first, but with a little practice, you'll be a pro in no time! Just remember to focus on the denominator, find the values that make it zero, and exclude those values from your domain. Keep practicing, and you'll master this concept! Remember, the domain is your friend! Understanding it is crucial for working with functions and understanding their behavior.

I hope this explanation was helpful. Happy calculating, and remember to always double-check your work!