Finding Real Values For Defined Expressions: A Detailed Guide

Hey guys! Today, we're diving into a crucial concept in mathematics: determining the set of real values for which expressions are defined. This is super important, especially when dealing with radicals (square roots, cube roots, etc.) and fractions. We'll break down each expression step-by-step, making sure you understand the logic behind each solution. So, grab your thinking caps, and let's get started!

Understanding the Basics: Why Does it Matter?

Before we jump into specific examples, let's quickly recap why we need to worry about the domain (the set of possible input values) of an expression. Some mathematical operations have restrictions. For example:

• Square roots: You can't take the square root of a negative number (in the realm of real numbers, at least). The expression under the square root (the radicand) must be greater than or equal to zero.
• Fractions: You can't divide by zero. The denominator of a fraction cannot be zero.

These restrictions are crucial because if we try to perform these operations on invalid inputs, the expression becomes undefined. Our goal is to find the values of x that avoid these undefined situations.

Problem Breakdown: Finding the Valid 'x' Values

Now, let's tackle each expression from your question. We'll go through them one by one, explaining the steps and the reasoning behind them. Remember, the key is to identify the restrictions imposed by the radicals and fractions and then solve the resulting inequalities or equations.

a) √(x-1)

In this expression, we have a square root. As we discussed, the expression under the square root (the radicand) must be greater than or equal to zero. So, we set up the following inequality:

x - 1 ≥ 0

To solve for x, we add 1 to both sides:

x ≥ 1

Therefore, the set of real values for which the expression √(x-1) is defined is x ≥ 1. In interval notation, this is written as [1, ∞).

b) √(√(2x-3))

This expression has a nested square root, meaning we have a square root inside another square root. This adds a layer of complexity, but the principle remains the same. We need to ensure that the expressions under both square roots are non-negative.

First, let's consider the inner square root: √(2x-3). The radicand must be greater than or equal to zero:

2x - 3 ≥ 0

Adding 3 to both sides gives:

2x ≥ 3

Dividing both sides by 2 gives:

x ≥ 3/2

Now, let's consider the outer square root. The entire expression inside the outer square root, which is √(2x-3), must also be non-negative. However, since the square root of any number is always non-negative (by definition), we only need to ensure that the inner radicand is non-negative, which we've already done.

Therefore, the set of real values for which the expression √(√(2x-3)) is defined is x ≥ 3/2. In interval notation, this is written as [3/2, ∞).

c) √(-5+12x)

Again, we have a square root, so the radicand must be greater than or equal to zero:

-5 + 12x ≥ 0

Adding 5 to both sides gives:

12x ≥ 5

Dividing both sides by 12 gives:

x ≥ 5/12

Therefore, the set of real values for which the expression √(-5+12x) is defined is x ≥ 5/12. In interval notation, this is written as [5/12, ∞).

d) √(-8-4x)

Same principle here! The radicand must be greater than or equal to zero:

-8 - 4x ≥ 0

Adding 8 to both sides gives:

-4x ≥ 8

Dividing both sides by -4 (and remember to flip the inequality sign because we're dividing by a negative number) gives:

x ≤ -2

Therefore, the set of real values for which the expression √(-8-4x) is defined is x ≤ -2. In interval notation, this is written as (-∞, -2].

e) 1/(√(8-2x))

This expression combines a square root and a fraction, so we have two restrictions to consider:

1. The radicand must be greater than or equal to zero:

   8 - 2x ≥ 0

2. The denominator cannot be zero:

   √(8 - 2x) ≠ 0

Let's solve the first inequality. Subtracting 8 from both sides gives:

-2x ≥ -8

Dividing both sides by -2 (and flipping the inequality sign) gives:

x ≤ 4

Now, let's address the second restriction. The square root cannot be zero, which means the radicand cannot be zero:

8 - 2x ≠ 0

Solving for x, we get:

x ≠ 4

Combining these two restrictions, we find that x must be less than 4, but it cannot be equal to 4.

Therefore, the set of real values for which the expression 1/(√(8-2x)) is defined is x < 4. In interval notation, this is written as (-∞, 4).

f) 3/√(3x-11)

Similar to the previous example, we have a fraction and a square root. We need to consider both restrictions:

1. The radicand must be greater than or equal to zero:

   3x - 11 ≥ 0

2. The denominator cannot be zero:

   √(3x - 11) ≠ 0

Solving the first inequality: 3x - 11 ≥ 0

Adding 11 to both sides gives:

3x ≥ 11

Dividing both sides by 3 gives:

x ≥ 11/3

Now, the second restriction: the radicand cannot be zero:

3x - 11 ≠ 0

Solving for x, we get:

x ≠ 11/3

Combining these, we see that x must be greater than 11/3, but it cannot be equal to 11/3.

Therefore, the set of real values for which the expression 3/√(3x-11) is defined is x > 11/3. In interval notation, this is written as (11/3, ∞).

h) 4/(-2x-7)

In this expression, we only have a fraction, so our sole restriction is that the denominator cannot be zero:

-2x - 7 ≠ 0

Adding 7 to both sides gives:

-2x ≠ 7

Dividing both sides by -2 gives:

x ≠ -7/2

Therefore, the set of real values for which the expression 4/(-2x-7) is defined is all real numbers except x = -7/2. In interval notation, this is written as (-∞, -7/2) ∪ (-7/2, ∞).

Key Takeaways: Mastering the Art of Finding Valid 'x' Values

Alright, we've tackled all the expressions! Let's recap the key principles we used to solve these problems:

1. Identify the Restrictions: Always start by pinpointing the restrictions imposed by the mathematical operations involved. Square roots require non-negative radicands, and fractions require non-zero denominators.
2. Set up Inequalities or Equations: Translate the restrictions into mathematical statements (inequalities or equations).
3. Solve for x: Use algebraic techniques to solve for x.
4. Consider Multiple Restrictions: If an expression has multiple restrictions (like fractions with square roots), you need to satisfy all of them.
5. Express the Solution: Write your answer clearly, whether in inequality notation or interval notation.

By mastering these steps, you'll be well-equipped to handle any problem that asks you to determine the set of real values for which an expression is defined. Remember, practice makes perfect, so keep working through examples!

Final Thoughts

Finding the domain of an expression might seem a bit abstract at first, but it's a fundamental skill in mathematics. It ensures that the operations we perform are valid and that our results are meaningful. So, keep practicing, and you'll become a pro in no time! You've got this, guys! Remember, understanding the limitations of mathematical operations is crucial for building a strong foundation in math. And hey, if you ever get stuck, don't hesitate to ask for help or review the basic principles. Happy problem-solving! This skill is essential for any aspiring mathematician or anyone who wants to use math in their daily life. So, let's keep sharpening our minds and exploring the world of mathematics! This is just the beginning, and there's so much more to discover!