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 $$ x=\square $$. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) 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 $$ x=\square $$. (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**

The given function is 
$$
f(x)=\frac{x^2+10x+24}{x^2+8x+16}.
$$
Factor the numerator and denominator:
- Numerator: 
  $$
  x^2+10x+24=(x+4)(x+6).
  $$
- Denominator: 
  $$
  x^2+8x+16=(x+4)^2.
  $$

Thus, 
$$
f(x)=\frac{(x+4)(x+6)}{(x+4)^2} \quad \text{for } x\neq -4.
$$
This simplifies to 
$$
f(x)=\frac{x+6}{x+4} \quad \text{with } x\neq -4.
$$

**Step 2. Find the Domain**

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

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

**Step 3. Find the $$x$$-intercept(s)**

The $$x$$-intercepts occur when $$f(x)=0$$. From the simplified form,
$$
\frac{x+6}{x+4}=0,
$$
the fraction is zero when the numerator equals zero (and the denominator is not zero). Set:
$$
x+6=0 \quad \Longrightarrow \quad x=-6.
$$
Since $$-6 \neq -4$$, this is valid.

**$$x$$-Intercept Answer:**  
A. The $$x$$-intercept(s) is/are at $$x=-6$$.

**Step 4. Find the $$y$$-intercept(s)**

The $$y$$-intercept occurs when $$x=0$$. Substitute $$x=0$$:
$$
f(0)=\frac{0+6}{0+4}=\frac{6}{4}=\frac{3}{2}.
$$

**$$y$$-Intercept Answer:**  
A. The $$y$$-intercept(s) is/are at $$y=\frac{3}{2}$$.

**Additional Graphing Information**

- **Vertical Asymptote:**  
  The function is undefined when $$x=-4$$ (because the denominator is zero), so there is a vertical asymptote at $$x=-4$$.

- **Horizontal Asymptote:**  
  Since the simplified function $$\frac{x+6}{x+4}$$ is a rational function of degree 1 in both numerator and denominator, the horizontal asymptote is the ratio of the leading coefficients:
  $$
  y= \frac{1}{1}=1.
  $$

- **$$x$$-Intercept:**  
  The graph crosses the $$x$$-axis at $$(-6,0)$$.

- **$$y$$-Intercept:**  
  The graph crosses the $$y$$-axis at $$(0,\frac{3}{2})$$.

These key elements summarize the graphing strategy for $$ y=f(x) $$.
**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:** $$ y = 1 $$. Graph: - Crosses the $$ x $$-axis at $$(-6, 0)$$. - Crosses the $$ y $$-axis at $$(0, \frac{3}{2})$$. - Has a vertical asymptote at $$ x = -4 $$ and a horizontal asymptote at $$ y = 1 $$.

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