Consider the following rational functions. $$ f(x)=\frac{3 x}{x^{2}-9} $$ $$ h(x)=\frac{3}{x^{2}-x-2} $$ Choose the graph of each function from the choices

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We start by analyzing each function separately.

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**1. Function $$\displaystyle f(x)=\frac{3x}{x^2-9}$$**

- **Factor the denominator:**

$$
  x^2-9=(x-3)(x+3)
  $$

The domain of $$ f(x) $$ excludes values that make the denominator zero. Thus, we exclude

$$
  x-3=0\quad\Rightarrow\quad x=3,\qquad x+3=0\quad\Rightarrow\quad x=-3.
  $$

So, $$ x\neq 3 $$ and $$ x\neq -3 $$.

- **Vertical asymptotes:**

Since there is no cancellation between factors in the numerator and denominator, $$ f(x) $$ has vertical asymptotes at

$$
  x = 3 \quad \text{and} \quad x = -3.
  $$

- **Horizontal asymptote:**

The degree of the numerator (1) is less than the degree of the denominator (2), so

$$
  y=0
  $$

is the horizontal asymptote.

- **Intercepts:**

- **$$x$$-intercept:** Set the numerator equal to zero:
    
    $$
    3x = 0\quad\Rightarrow\quad x=0.
    $$
    
    Since $$ 0 $$ is in the domain (denom $$=0^2-9=-9\neq0$$), the function passes through
    
    $$
    (0, 0).
    $$

- **$$y$$-intercept:** Evaluated by setting $$ x=0 $$ (which we just did), giving
    
    $$
    f(0)=0.
    $$

- **Symmetry:**

Check whether the function is even or odd by substituting $$-x$$:

$$
  f(-x)=\frac{3(-x)}{(-x)^2-9}=\frac{-3x}{x^2-9}=-f(x).
  $$

Therefore, $$ f(x) $$ is an odd function (symmetric about the origin).

- **Graph summary for $$ f(x) $$:**

- Vertical asymptotes: $$ x = 3 $$ and $$ x = -3 $$
  - Horizontal asymptote: $$ y = 0 $$
  - $$ x$$-intercept: $$ (0, 0) $$ (also the $$y$$-intercept)
  - Odd symmetry

---

**2. Function $$\displaystyle h(x)=\frac{3}{x^2-x-2}$$**

- **Factor the denominator:**

Factor the quadratic:

$$
  x^2-x-2=(x-2)(x+1)
  $$

The values that make the denominator zero are

$$
  x-2=0\quad\Rightarrow\quad x=2,\qquad x+1=0\quad\Rightarrow\quad x=-1.
  $$

So, the domain excludes $$ x=2 $$ and $$ x=-1 $$.

- **Vertical asymptotes:**

There is no cancellation with the numerator, so the vertical asymptotes are at

$$
  x = 2 \quad \text{and} \quad x = -1.
  $$

- **Horizontal asymptote:**

Since the degree of the numerator (0, since $$3$$ is constant) is less than the degree of the denominator (2), we have

$$
  y = 0.
  $$

- **Intercepts:**

- **$$x$$-intercept:** The numerator is $$3$$ (which is never zero), so there is no $$x$$-intercept.
  
  - **$$y$$-intercept:** Set $$ x=0 $$:
    
    $$
    h(0)=\frac{3}{0^2-0-2}=\frac{3}{-2}=-\frac{3}{2}.
    $$
    
    So the $$y$$-intercept is
    
    $$
    \left(0,-\frac{3}{2}\right).
    $$

- **Symmetry:**

Testing for symmetry by replacing $$ x $$ with $$-x$$ does not yield a relationship like $$ h(-x)=h(x) $$ or $$ h(-x)=-h(x) $$; hence, $$ h(x) $$ is not even or odd.

- **Graph summary for $$ h(x) $$:**

- Vertical asymptotes: $$ x = 2 $$ and $$ x = -1 $$
  - Horizontal asymptote: $$ y = 0 $$
  - $$y$$-intercept: $$ \left(0, -\frac{3}{2}\right) $$
  - No $$x$$-intercepts

---

**Conclusion:**

- The graph of $$ f(x)=\frac{3x}{x^2-9} $$ has vertical asymptotes at $$ x=-3 $$ and $$ x=3 $$, a horizontal asymptote at $$ y=0 $$, passes through the origin, and is symmetric about the origin (odd function).

- The graph of $$ h(x)=\frac{3}{x^2-x-2} $$ has vertical asymptotes at $$ x=-1 $$ and $$ x=2 $$, a horizontal asymptote at $$ y=0 $$, a $$y$$-intercept at $$ \left(0,-\frac{3}{2}\right) $$, and it does not cross the $$x$$-axis.

When choosing the graphs from the options provided, look for these characteristics in each graph.
The graph of $$ f(x) = \frac{3x}{x^2 - 9} $$ has vertical asymptotes at $$ x = -3 $$ and $$ x = 3 $$, a horizontal asymptote at $$ y = 0 $$, passes through the origin, and is symmetric about the origin. The graph of $$ h(x) = \frac{3}{x^2 - x - 2} $$ has vertical asymptotes at $$ x = -1 $$ and $$ x = 2 $$, a horizontal asymptote at $$ y = 0 $$, a $$ y $$-intercept at $$ (0, -\frac{3}{2}) $$, and no $$ x $$-intercepts.

Consider the following rational functions. \[ f(x)=\frac{3 x}{x^{2}-9} \] \( h(x)=\frac{3}{x^{2}-x-2} \) Choose the graph of each function from the choices

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