Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of $$ y=f(x) $$. $$ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} $$ Find the domain of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is all real $$ x $$, except $$ x= $$ B. The domain is all real $$ x $$. Find the $$ x $$-intercepts of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $$ x $$-intercept(s) is/are at $$ x= $$ (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no $$ x $$-intercepts. Find the $$ y $$-intercepts of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $$ y $$-intercept(s) is/are at $$ y=\square $$. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no $$ y $$-intercepts.

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**Step 1. Factor the Function**

Given
$$
f(x)=\frac{x^{2}+10x+24}{x^{2}+8x+16},
$$
we factor both the numerator and denominator.

For the numerator:
$$
x^2+10x+24=(x+4)(x+6).
$$

For the denominator:
$$
x^2+8x+16=(x+4)^2.
$$

Thus, for $$ x\neq -4 $$ (since the factor $$ x+4 $$ causes the denominator to be 0),
$$
f(x)=\frac{(x+4)(x+6)}{(x+4)^2}=\frac{x+6}{x+4}.
$$

---

**Step 2. Determine the Domain**

The denominator of the original function,
$$
(x+4)^2,
$$
is zero when 
$$
x+4=0 \quad \Longrightarrow \quad x=-4.
$$
Thus, the domain is all real numbers except $$ x=-4 $$.

*Answer:*  
A. The domain is all real $$ x $$, except $$ x= -4 $$.

---

**Step 3. Find the $$ x $$-Intercepts**

The $$ x $$-intercepts occur when $$ f(x)=0 $$. From the simplified form,
$$
\frac{x+6}{x+4}=0,
$$
the numerator must be zero (and the denominator must not be zero). Setting the numerator equal to zero gives:
$$
x+6=0 \quad \Longrightarrow \quad x=-6.
$$

Since $$ -6\neq -4 $$, this is valid.

*Answer:*  
A. The $$ x $$-intercept is at $$ x=-6 $$ (or the point $$(-6,0)$$).

---

**Step 4. Find the $$ y $$-Intercepts**

The $$ y $$-intercept is found by evaluating $$ f(0) $$. Substitute $$ x=0 $$ into the simplified form:
$$
f(0)=\frac{0+6}{0+4}=\frac{6}{4}=\frac{3}{2}.
$$

*Answer:*  
A. The $$ y $$-intercept is at $$ y=\frac{3}{2} $$ (or the point $$(0,\frac{3}{2})$$).

---

**Additional Graphing Information**

1. **Vertical Asymptote:**  
   Since the denominator $$ x+4 $$ is zero at $$ x=-4 $$ and is not canceled, there is a vertical asymptote at  
   $$
   x=-4.
   $$

2. **Horizontal Asymptote:**  
   The degrees of the numerator and denominator (in the simplified form) are both 1. Thus, the horizontal asymptote is given by the ratio of the leading coefficients:
   $$
   y=\frac{1}{1}=1.
   $$

3. **Summary for Graphing:**  
   - The function simplifies to $$ f(x)=\frac{x+6}{x+4} $$ for $$ x\neq -4 $$.
   - **Domain:** $$ x\in\mathbb{R} \setminus \{-4\} $$.
   - **$$ x $$-Intercept:** $$ x=-6 $$ (point $$(-6,0)$$).
   - **$$ y $$-Intercept:** $$ y=\frac{3}{2} $$ (point $$(0,\frac{3}{2})$$).
   - **Vertical Asymptote:** $$ x=-4 $$.
   - **Horizontal Asymptote:** $$ y=1 $$.

These details provide a comprehensive view needed to sketch the graph of $$ y=f(x) $$.
**Summary of $$ f(x) = \frac{x^{2}+10x+24}{x^{2}+8x+16} $$:** - **Domain:** All real numbers except $$ x = -4 $$. - **$$ x $$-Intercept:** $$ x = -6 $$. - **$$ y $$-Intercept:** $$ y = \frac{3}{2} $$. - **Vertical Asymptote:** $$ x = -4 $$. - **Horizontal Asymptote:** $$ y = 1 $$. These points help in sketching the graph of $$ y = f(x) $$.

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