Answer :
Sure! Let's solve the equation [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex] step-by-step.
### Step 1: Understand the Structure
The function given is [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex]. This is a fractional function where the numerator is 1 and the denominator is the square root of [tex]\( x^2 - 4 \)[/tex].
### Step 2: Identify Domain Constraints
Before diving into any deeper analysis, we need to consider the domain of the function. The value inside the square root, [tex]\( x^2 - 4 \)[/tex], must be positive because the square root of a negative number is not defined in the real numbers, and it must not be zero to avoid division by zero.
So, we need [tex]\( x^2 - 4 > 0 \)[/tex]:
#### Solve [tex]\( x^2 - 4 > 0 \)[/tex]:
- [tex]\( x^2 > 4 \)[/tex]
- [tex]\( x > 2 \)[/tex] or [tex]\( x < -2 \)[/tex]
Thus, the domain of the function is [tex]\( x \in (-\infty, -2) \cup (2, \infty) \)[/tex].
### Step 3: Interpretation of the Function
The function [tex]\( y \)[/tex] will take on real values only within its domain. For any [tex]\( x \)[/tex] in the intervals [tex]\( (-\infty, -2) \cup (2, \infty) \)[/tex]:
- The quantity [tex]\( x^2 - 4 \)[/tex] is always positive.
- The square root of a positive number is real.
- Therefore, the fraction [tex]\( \frac{1}{\sqrt{x^2 - 4}} \)[/tex] is well-defined.
### Step 4: Graphical Understanding
To understand the behavior of [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex]:
1. For large values of [tex]\( x \)[/tex]:
- As [tex]\( x \)[/tex] moves further from 2 and -2, [tex]\( x^2 - 4 \)[/tex] increases, making [tex]\( \sqrt{x^2 - 4} \)[/tex] larger.
- Hence, [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex] will decrease towards zero.
2. For [tex]\( x \)[/tex] near ±2:
- As [tex]\( x \)[/tex] approaches 2 from the right or -2 from the left, [tex]\( x^2 - 4 \)[/tex] approaches zero.
- Since [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex], as [tex]\( x \)[/tex] gets closer to ±2, the denominator approaches zero, making [tex]\( y \)[/tex] approach infinity.
### Step 5: Vertical Asymptotes
- The function has vertical asymptotes at [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex], because as [tex]\( x \)[/tex] approaches these values, the function value [tex]\( y \)[/tex] tends to infinity.
### Step 6: Horizontal Asymptote
- As [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex], [tex]\( y \)[/tex] tends to [tex]\( 0 \)[/tex]. This introduces a horizontal asymptote at [tex]\( y = 0 \)[/tex].
### Conclusion
The function [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex]:
- Is defined for [tex]\( x \in (-\infty, -2) \cup (2, \infty) \)[/tex].
- Has vertical asymptotes at [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
- Has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
This detailed analysis covers the understanding of the function structure, domain constraints, and graphical behavior.
### Step 1: Understand the Structure
The function given is [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex]. This is a fractional function where the numerator is 1 and the denominator is the square root of [tex]\( x^2 - 4 \)[/tex].
### Step 2: Identify Domain Constraints
Before diving into any deeper analysis, we need to consider the domain of the function. The value inside the square root, [tex]\( x^2 - 4 \)[/tex], must be positive because the square root of a negative number is not defined in the real numbers, and it must not be zero to avoid division by zero.
So, we need [tex]\( x^2 - 4 > 0 \)[/tex]:
#### Solve [tex]\( x^2 - 4 > 0 \)[/tex]:
- [tex]\( x^2 > 4 \)[/tex]
- [tex]\( x > 2 \)[/tex] or [tex]\( x < -2 \)[/tex]
Thus, the domain of the function is [tex]\( x \in (-\infty, -2) \cup (2, \infty) \)[/tex].
### Step 3: Interpretation of the Function
The function [tex]\( y \)[/tex] will take on real values only within its domain. For any [tex]\( x \)[/tex] in the intervals [tex]\( (-\infty, -2) \cup (2, \infty) \)[/tex]:
- The quantity [tex]\( x^2 - 4 \)[/tex] is always positive.
- The square root of a positive number is real.
- Therefore, the fraction [tex]\( \frac{1}{\sqrt{x^2 - 4}} \)[/tex] is well-defined.
### Step 4: Graphical Understanding
To understand the behavior of [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex]:
1. For large values of [tex]\( x \)[/tex]:
- As [tex]\( x \)[/tex] moves further from 2 and -2, [tex]\( x^2 - 4 \)[/tex] increases, making [tex]\( \sqrt{x^2 - 4} \)[/tex] larger.
- Hence, [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex] will decrease towards zero.
2. For [tex]\( x \)[/tex] near ±2:
- As [tex]\( x \)[/tex] approaches 2 from the right or -2 from the left, [tex]\( x^2 - 4 \)[/tex] approaches zero.
- Since [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex], as [tex]\( x \)[/tex] gets closer to ±2, the denominator approaches zero, making [tex]\( y \)[/tex] approach infinity.
### Step 5: Vertical Asymptotes
- The function has vertical asymptotes at [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex], because as [tex]\( x \)[/tex] approaches these values, the function value [tex]\( y \)[/tex] tends to infinity.
### Step 6: Horizontal Asymptote
- As [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex], [tex]\( y \)[/tex] tends to [tex]\( 0 \)[/tex]. This introduces a horizontal asymptote at [tex]\( y = 0 \)[/tex].
### Conclusion
The function [tex]\( y = \frac{1}{\sqrt{x^2 - 4}} \)[/tex]:
- Is defined for [tex]\( x \in (-\infty, -2) \cup (2, \infty) \)[/tex].
- Has vertical asymptotes at [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
- Has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
This detailed analysis covers the understanding of the function structure, domain constraints, and graphical behavior.
