Alright class, let's dive into the solution for the given problem.
We are asked to analyze the function:
[tex]\[ y = \frac{1}{\sqrt{(3 + x)(7 + x)}} \][/tex]
Here’s how we can comprehend this function step-by-step:
### Step 1: Understanding the function
We want to understand the behavior of the function:
[tex]\[ y = \frac{1}{\sqrt{(3 + x)(7 + x)}} \][/tex]
Our goal is to ensure that the function is well-defined.
### Step 2: Identify the Domain
The expression inside the square root, [tex]\((3 + x)(7 + x)\)[/tex], must be positive (greater than zero) since the square root function is defined only for non-negative values and we cannot divide by zero.
Domain Consideration:
[tex]\[ (3 + x)(7 + x) > 0 \][/tex]
### Step 3: Equality Conditions
First, we find the points where the expression inside the square root is equal to zero. That will help us identify the regions wherein the expression is positive or negative.
[tex]\[ (3 + x)(7 + x) = 0 \][/tex]
[tex]\[
(3 + x) = 0 \quad \text{or} \quad (7 + x) = 0
\][/tex]
[tex]\[ x = -3 \quad \text{or} \quad x = -7 \][/tex]
These points split the number line into three intervals:
[tex]\[ (-\infty, -7), \quad (-7, -3), \quad (-3, \infty) \][/tex]
### Step 4: Test the Intervals
To determine the sign of [tex]\((3 + x)(7 + x)\)[/tex] in each interval, choose a test point from each interval and evaluate the expression.
1. Interval [tex]\((- \infty, -7)\)[/tex]:
- Test Point: [tex]\( x = -8 \)[/tex]
- Evaluate:
[tex]\[ (3 + (-8))(7 + (-8)) = (-5)(-1) = 5 \][/tex]
Positive
2. Interval [tex]\((-7, -3)\)[/tex]:
- Test Point: [tex]\( x = -5 \)[/tex]
- Evaluate:
[tex]\[ (3 + (-5))(7 + (-5)) = (-2)(2) = -4 \][/tex]
Negative
3. Interval [tex]\((-3, \infty)\)[/tex]:
- Test Point: [tex]\( x = 0 \)[/tex]
- Evaluate:
[tex]\[ (3)(7) = 21 \][/tex]
Positive
To avoid undefined behavior, we exclude the intervals where the expression is negative or zero.
### Step 5: Conclusion of the Domain
From above, the function is defined for:
[tex]\[ x \in (-\infty, -7) \cup (-3, \infty) \][/tex]
### Step 6: Simplify the Expression
Given:
[tex]\[ y = \frac{1}{\sqrt{(3 + x)(7 + x)}} \][/tex]
This expression cannot be simplified further without altering the structure of the given conditions. This is already a simplified form containing all necessary components.
### Step 7: Final Expression
Thus, the function can be summarized as:
[tex]\[ y = \frac{1}{\sqrt{(3 + x)(7 + x)}} \][/tex]
for:
[tex]\[ x \in (-\infty, -7) \cup (-3, \infty) \][/tex]
This completes the detailed step-by-step solution for understanding the given function. Any questions regarding any steps?
