Finding The Domain Of A Function: A Step-by-Step Guide

Oct 17, 2025 by SLV Team 55 views

Hey guys! Today, we're going to dive into a math problem that might seem a bit tricky at first, but trust me, we'll break it down step by step. We'll be figuring out the domain of the function ${ f(x) = \sqrt{\frac{x^2+2x-3}{x-4}} }$ . Understanding how to find the domain is super important in math because it tells us all the possible x-values that we can plug into our function without running into any issues like division by zero or taking the square root of a negative number. This concept is fundamental, so let's get started!

Understanding the Basics: What is a Domain?

Before we jump into the problem, let's make sure we're all on the same page about what the domain of a function actually is. The domain is simply the set of all possible input values (x-values) for which the function is defined. Think of it as the set of numbers that you're allowed to plug into the function. For some functions, the domain is all real numbers (any number is fine!), but for others, there might be restrictions. These restrictions usually come from two main things: avoiding division by zero and avoiding the square root of a negative number (which gives us imaginary numbers, and we're sticking with real numbers here).

In our case, we have a square root and a fraction, so we'll need to consider both of these rules to find the domain. The function ${ f(x) = \sqrt{\frac{x^2+2x-3}{x-4}} }$ is a combination of a square root function and a rational function. Therefore, the domain of the function is determined by the following two conditions:

The expression inside the square root must be greater than or equal to zero (because we can't take the square root of a negative number).
The denominator of the fraction cannot be equal to zero (because we can't divide by zero).

So, finding the domain is all about identifying those x-values that work and excluding those that don't. Let's get our hands dirty and start solving the problem!

Breaking Down the Problem: Step-by-Step Solution

Alright, let's get down to the nitty-gritty of solving this problem. To find the domain, we need to satisfy two conditions, as we mentioned earlier. Let's tackle these one at a time.

Condition 1: The Expression Inside the Square Root

The first condition is that the expression inside the square root, ${\frac{x^2+2x-3}{x-4}}$ , must be greater than or equal to zero. This means we need to solve the inequality: ${\frac{x^2+2x-3}{x-4} \geq 0}$ . To solve this inequality, we'll first factor the numerator and find the critical points.

Factor the Numerator: The quadratic expression ${x^2 + 2x - 3}$ can be factored into ${(x+3)(x-1)}$ . So, our inequality becomes: ${\frac{(x+3)(x-1)}{x-4} \geq 0}$ .
Find Critical Points: The critical points are the values of x that make the numerator or the denominator equal to zero. From the factored form, we get the following critical points: ${x = -3}$ , ${x = 1}$ , and ${x = 4}$ (from the denominator).
Create a Sign Chart: A sign chart helps us determine the intervals where the expression is positive or negative. We'll mark the critical points on a number line and test the expression in the intervals created by these points.
- Interval 1: ${x < -3}$ – Choose a test value, say ${x = -4}$ . Plug this into ${\frac{(x+3)(x-1)}{x-4}}$ . We get ${\frac{(-1)(-5)}{-8}}$ , which is negative.
- Interval 2: ${-3 < x < 1}$ – Choose ${x = 0}$ . We get ${\frac{(3)(-1)}{-4}}$ , which is positive.
- Interval 3: ${1 < x < 4}$ – Choose ${x = 2}$ . We get ${\frac{(5)(1)}{-2}}$ , which is negative.
- Interval 4: ${x > 4}$ – Choose ${x = 5}$ . We get ${\frac{(8)(4)}{1}}$ , which is positive.
Determine the Solution: The inequality ${\frac{(x+3)(x-1)}{x-4} \geq 0}$ is satisfied when the expression is positive or equal to zero. From our sign chart, the solution includes the intervals where the expression is positive, plus the points where the numerator is zero. So, the solution to this inequality is ${x \leq -3}$ or ${1 \leq x < 4}$ or ${x > 4}$ . Note that we use ${x < 4}$ because the denominator can't be zero.

Condition 2: The Denominator Cannot Be Zero

The second condition is that the denominator, ${x - 4}$ , cannot be equal to zero. This means ${x \neq 4}$ . We've already taken this into account when we used ${x < 4}$ and ${x > 4}$ in our solution of Condition 1.

Combining the Solutions

Now, let's put it all together! We have the solution for Condition 1: ${x \leq -3}$ or ${1 \leq x < 4}$ or ${x > 4}$ . Since we've already accounted for the restriction in the denominator, this is our final solution for the domain of the function. Therefore, the domain of the function ${f(x) = \sqrt{\frac{x^2+2x-3}{x-4}} }$ is ${x \leq -3}$ or ${1 \leq x < 4}$ or ${x > 4}$ . Now, let's look at the multiple-choice options and see which one matches our solution.

Matching the Answer Choice: Identifying the Correct Domain

Okay, we've done the hard work, and now it's time to find the correct answer from the choices given. We found that the domain of the function is all real numbers such that ${x \leq -3}$ or ${1 \leq x < 4}$ or ${x > 4}$ . Let's compare this with the answer choices:

(A) {x|1 ≤x<4, x ∈ R}: This is incorrect because it only includes a portion of the domain.
**(B) x | x ≤ -1 atau 3 < x < 4, x ∈R}** This is incorrect because the inequality is wrong. The interval should include ${x \leq -3$, not ${x \leq -1}$ .
**(C) x | x ≤ -3 atau 1 < x < 4, x ∈ R }** This is incorrect. Though it correctly identifies ${x \leq -3$, it states ${1 < x < 4}$ , the correct answer must include the value of 1. It should be ${1 \leq x < 4}$ .
(D) {x|1≤x≤ 3 atau x > 4, x ∈R}: This is incorrect. The solution must include values less than or equal to -3.
(E) {x|-3 ≤ x ≤ 1 atau x >...: This is incorrect as it presents the incorrect inequality interval, and it is also incomplete.

Looking back at our solution, we see that none of the provided options exactly match our derived domain. It is crucial to be accurate in identifying the correct domain of a function to ensure that it's correctly understood. Always double-check your work and consider how each part of the function affects the domain. So, we can conclude that the given options do not have the proper solution. This might happen due to a typo or mistake in the provided options. Therefore, there is no correct answer from the provided multiple choices.

Conclusion: Mastering the Domain

Congrats, guys! You've successfully navigated the process of finding the domain of a function that involves a square root and a fraction. Remember, the key is to break down the problem into smaller steps, consider all restrictions (no division by zero and no square root of a negative number), and use tools like sign charts to help you visualize the solution.

Finding the domain can seem intimidating at first, but with practice, it becomes much easier. Keep practicing, and you'll become a pro at identifying the domains of all sorts of functions. Keep in mind that understanding domains is fundamental to working with functions, and it will help you in other areas of mathematics as well. If you have any questions or want to try another example, just let me know. Happy learning, and see you next time!