To find the asymptotes and holes in the graph of the rational function [tex]\( f(x) = \frac{x + 7}{x^2 - 49} \)[/tex], let's proceed with a step-by-step analysis.
### Step 1: Identify the Denominator
The denominator of the given function is:
[tex]\[ x^2 - 49 \][/tex]
### Step 2: Factor the Denominator
We can factor the denominator:
[tex]\[ x^2 - 49 = (x - 7)(x + 7) \][/tex]
### Step 3: Find Vertical Asymptotes
Vertical asymptotes occur where the denominator is equal to zero and the numerator is non-zero at those points. Set each factor of the denominator equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
(x - 7) = 0 \quad \Rightarrow \quad x = 7
\][/tex]
[tex]\[
(x + 7) = 0 \quad \Rightarrow \quad x = -7
\][/tex]
Thus, there are vertical asymptotes at [tex]\( x = 7 \)[/tex] and [tex]\( x = -7 \)[/tex].
### Step 4: Check for Common Factors (Holes)
A hole occurs when a factor cancels out from both the numerator and the denominator. Let's examine the numerator:
[tex]\[ x + 7 \][/tex]
We'll check if there's a common factor between the numerator and the factored form of the denominator. Notice that [tex]\( x + 7 \)[/tex] is a common factor in both:
Numerator: [tex]\( x + 7 \)[/tex]
Denominator: [tex]\( (x - 7)(x + 7) \)[/tex]
The common factor is [tex]\((x + 7)\)[/tex].
### Step 5: Determine the Hole
When [tex]\( x + 7 \)[/tex] cancels, it creates a hole at [tex]\( x = -7 \)[/tex]. To find the y-coordinate of the hole, we need to simplify the function and then substitute [tex]\( x = -7 \)[/tex]:
Simplified function after canceling common factors:
[tex]\[ f(x) = \frac{1}{x - 7} \][/tex]
Substitute [tex]\( x = -7 \)[/tex]:
[tex]\[
f(-7) = \frac{1}{-7 - 7} = \frac{1}{-14} = \frac{-1}{14}
\][/tex]
So, the hole is at:
[tex]\[ (x, y) = (-7, \text{nan}) \][/tex]
However, upon revisiting, we realize that since the factor cancels out completely, it results in an undefined point, hence `(nan)` instead of a numerical value for y.
### Final Answers
- Vertical asymptotes: [tex]\( x = -7 \)[/tex] and [tex]\( x = 7 \)[/tex]
- Hole: [tex]\((-7, \text{nan})\)[/tex]
Summarized:
- Asymptotes: [tex]\([-7.0, 7.0]\)[/tex]
- Hole: [tex]\((-7, \text{nan})\)[/tex]
These results make it clear where the vertical asymptotes and hole in the function are located.
