Find the domain and the $$ x $$ - and $$ y $$-intercepts of $$ f(x) $$. Then graph the function. $$ f(x)=\frac{x-2}{(x+4)^{2}} $$ What is the domain of $$ f(x) $$ ? A. $$ (-\infty, 2) \cup(2, \infty) $$ B. $$ (-\infty, \infty) $$ Celect the correct choice below and fill in any answer boxes within your choice. A. The x-intercept is (Type an ordered pair. Type an exact answer.) B. There is no $$ x $$-intercept.

Question

UpStudy AI · Accepted Answer

**Step 1. Find the Domain**

The function is  
$$
f(x)=\frac{x-2}{(x+4)^2}.
$$
The denominator $$(x+4)^2$$ is zero when  
$$
x+4=0 \quad \Longrightarrow \quad x=-4.
$$
Thus, $$ x = -4 $$ must be excluded from the domain. The domain is all real numbers except $$-4$$:

$$
\text{Domain: } (-\infty, -4) \cup (-4, \infty).
$$

**Step 2. Find the $$ x $$-Intercept**

The $$ x $$-intercept occurs when $$ f(x)=0 $$. Set the numerator equal to zero:

$$
x-2=0 \quad \Longrightarrow \quad x=2.
$$

Since $$ 2 \neq -4 $$ (the excluded value), the $$ x $$-intercept is valid:

$$
\text{$$ x $$-intercept: } (2,0).
$$

**Step 3. Find the $$ y $$-Intercept**

The $$ y $$-intercept is found by evaluating $$ f(0) $$:

$$
f(0)=\frac{0-2}{(0+4)^2}=\frac{-2}{16}=-\frac{1}{8}.
$$

Thus, the $$ y $$-intercept is:

$$
(0,-\frac{1}{8}).
$$

**Step 4. Graphing the Function**

Key features to include in the graph:

- **Vertical Asymptote:** $$ x=-4 $$, since the denominator is zero at $$ x=-4 $$.
- **Horizontal Asymptote:** Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is $$ y=0 $$.
- **Intercepts:** $$ x $$-intercept at $$ (2,0) $$ and $$ y $$-intercept at $$ (0,-\frac{1}{8}) $$.

When graphing, plot the intercepts, draw the vertical line $$ x=-4 $$ as an asymptote, and indicate that the graph approaches the line $$ y=0 $$ as $$ x \to \pm \infty $$.

**Answering the Multiple-Choice Parts:**

- The correct domain is **$$ (-\infty, -4) \cup (-4, \infty) $$**.
- For the $$ x $$-intercept, the correct choice is:  
  **A. The $$ x $$-intercept is $$ (2,0) $$.**
**Domain:** All real numbers except $$ x = -4 $$. **$$ x $$-Intercept:** $$ (2, 0) $$. **$$ y $$-Intercept:** $$ (0, -\frac{1}{8}) $$.

Find the domain and the - and -intercepts of . Then graph the function.

What is the domain of ?
A.
B.
Celect the correct choice below and fill in any answer boxes within your choice.
A. The x-intercept is
(Type an ordered pair. Type an exact answer.)
B. There is no -intercept.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Calculus Questions

Find the domain and the - and -intercepts of . Then graph the function. What is the domain of ? A. B. Celect the correct choice below and fill in any answer boxes within your choice. A. The x-intercept is (Type an ordered pair. Type an exact answer.) B. There is no -intercept.