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To determine which graph represents the function [tex]\(f(x) = \frac{5 - 5x^2}{x^2}\)[/tex], let's analyze the behavior and key characteristics of the function step by step.
### Step 1: Simplify the Function
First, let's simplify the given function:
[tex]\[ f(x) = \frac{5 - 5x^2}{x^2} \][/tex]
We can factor out a common factor from the numerator:
[tex]\[ f(x) = \frac{5(1 - x^2)}{x^2} \][/tex]
Next, we can separate the terms in the fraction:
[tex]\[ f(x) = 5 \left( \frac{1}{x^2} - 1 \right) \][/tex]
This simplifies to:
[tex]\[ f(x) = 5 \left( \frac{1 - x^2}{x^2} \right) = 5 \left( \frac{1}{x^2} - 1 \right) \][/tex]
[tex]\[ f(x) = \frac{5}{x^2} - 5 \][/tex]
### Step 2: Determine Key Characteristics
Now, let’s determine the key characteristics of the function based on its simplified form:
#### i. Vertical Asymptotes
The function will have a vertical asymptote where [tex]\(x^2 = 0\)[/tex], which occurs at [tex]\(x = 0\)[/tex]. So, there is a vertical asymptote at [tex]\(x = 0\)[/tex].
#### ii. Horizontal Asymptote
As [tex]\(x\)[/tex] approaches [tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex], [tex]\(\frac{5}{x^2} \)[/tex] approaches 0. Therefore, the horizontal asymptote is at [tex]\(y = -5\)[/tex].
#### iii. Intercepts
- y-intercept: The function is undefined at [tex]\(x = 0\)[/tex], so there is no y-intercept.
- x-intercepts: Set [tex]\(f(x) = 0\)[/tex]:
[tex]\[ \frac{5}{x^2} - 5 = 0 \][/tex]
[tex]\[ \frac{5}{x^2} = 5 \][/tex]
[tex]\[ \frac{1}{x^2} = 1 \][/tex]
[tex]\[ x^2 = 1 \][/tex]
Therefore, [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex] are the x-intercepts.
### Step 3: Behavior Analysis
- For [tex]\(x > 1\)[/tex] or [tex]\(x < -1\)[/tex], [tex]\(\frac{5}{x^2}\)[/tex] is positive and less than 5, so [tex]\(f(x)\)[/tex] is negative and approaches -5.
- For [tex]\(0 < |x| < 1\)[/tex], [tex]\(\frac{5}{x^2}\)[/tex] is greater than 5, so [tex]\(f(x)\)[/tex] is positive.
- As [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex] from either side, [tex]\(\frac{5}{x^2}\to \infty\)[/tex], and thus [tex]\(f(x) \to \infty\)[/tex].
### Step 4: Graph Representation
Combining all the characteristics:
- The graph has vertical asymptote at [tex]\(x = 0\)[/tex].
- The graph has a horizontal asymptote at [tex]\(y = -5\)[/tex].
- The graph intersects the x-axis at [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex].
- The function is positive between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] (excluding [tex]\(0\)[/tex]) and negative outside that range.
By analyzing these characteristics, we can identify the correct graph representation of the function [tex]\(f(x) = \frac{5 - 5x^2}{x^2}\)[/tex]. The correct graph will show the described behavior around the key points and asymptotes.
### Step 1: Simplify the Function
First, let's simplify the given function:
[tex]\[ f(x) = \frac{5 - 5x^2}{x^2} \][/tex]
We can factor out a common factor from the numerator:
[tex]\[ f(x) = \frac{5(1 - x^2)}{x^2} \][/tex]
Next, we can separate the terms in the fraction:
[tex]\[ f(x) = 5 \left( \frac{1}{x^2} - 1 \right) \][/tex]
This simplifies to:
[tex]\[ f(x) = 5 \left( \frac{1 - x^2}{x^2} \right) = 5 \left( \frac{1}{x^2} - 1 \right) \][/tex]
[tex]\[ f(x) = \frac{5}{x^2} - 5 \][/tex]
### Step 2: Determine Key Characteristics
Now, let’s determine the key characteristics of the function based on its simplified form:
#### i. Vertical Asymptotes
The function will have a vertical asymptote where [tex]\(x^2 = 0\)[/tex], which occurs at [tex]\(x = 0\)[/tex]. So, there is a vertical asymptote at [tex]\(x = 0\)[/tex].
#### ii. Horizontal Asymptote
As [tex]\(x\)[/tex] approaches [tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex], [tex]\(\frac{5}{x^2} \)[/tex] approaches 0. Therefore, the horizontal asymptote is at [tex]\(y = -5\)[/tex].
#### iii. Intercepts
- y-intercept: The function is undefined at [tex]\(x = 0\)[/tex], so there is no y-intercept.
- x-intercepts: Set [tex]\(f(x) = 0\)[/tex]:
[tex]\[ \frac{5}{x^2} - 5 = 0 \][/tex]
[tex]\[ \frac{5}{x^2} = 5 \][/tex]
[tex]\[ \frac{1}{x^2} = 1 \][/tex]
[tex]\[ x^2 = 1 \][/tex]
Therefore, [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex] are the x-intercepts.
### Step 3: Behavior Analysis
- For [tex]\(x > 1\)[/tex] or [tex]\(x < -1\)[/tex], [tex]\(\frac{5}{x^2}\)[/tex] is positive and less than 5, so [tex]\(f(x)\)[/tex] is negative and approaches -5.
- For [tex]\(0 < |x| < 1\)[/tex], [tex]\(\frac{5}{x^2}\)[/tex] is greater than 5, so [tex]\(f(x)\)[/tex] is positive.
- As [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex] from either side, [tex]\(\frac{5}{x^2}\to \infty\)[/tex], and thus [tex]\(f(x) \to \infty\)[/tex].
### Step 4: Graph Representation
Combining all the characteristics:
- The graph has vertical asymptote at [tex]\(x = 0\)[/tex].
- The graph has a horizontal asymptote at [tex]\(y = -5\)[/tex].
- The graph intersects the x-axis at [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex].
- The function is positive between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] (excluding [tex]\(0\)[/tex]) and negative outside that range.
By analyzing these characteristics, we can identify the correct graph representation of the function [tex]\(f(x) = \frac{5 - 5x^2}{x^2}\)[/tex]. The correct graph will show the described behavior around the key points and asymptotes.
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