Plot all of the existing five features of the following
If you get a fraction or decimal then plot as close to

Question

Plot all of the existing five features of the following If you get a fraction or decimal then plot as close to $$ f(x)=\frac{x+3}{2 x-7} $$

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1. **Domain**

The function is defined for all $$ x $$ such that the denominator is not zero. Set the denominator equal to zero and solve:
   $$
   2x - 7 = 0 \quad \Rightarrow \quad x = \frac{7}{2}.
   $$
   Therefore, the domain is
   $$
   \{ x \in \mathbb{R} \mid x \neq \frac{7}{2} \}.
   $$

2. **x-intercept**

To find the $$ x $$-intercept, set $$ f(x) = 0 $$. This happens when the numerator is zero (provided the denominator is not zero):
   $$
   x+3=0 \quad \Rightarrow \quad x=-3.
   $$
   The $$ x $$-intercept is at
   $$
   (-3,0).
   $$

3. **y-intercept**

To find the $$ y $$-intercept, substitute $$ x=0 $$ into the function:
   $$
   f(0)=\frac{0+3}{2\cdot 0-7}=\frac{3}{-7}=-\frac{3}{7}.
   $$
   The $$ y $$-intercept is at
   $$
   \left(0,-\frac{3}{7}\right).
   $$

4. **Vertical Asymptote**

A vertical asymptote occurs where the denominator is zero (and the numerator is nonzero). We already found the zero of the denominator:
   $$
   x=\frac{7}{2}.
   $$
   Thus, there is a vertical asymptote at
   $$
   x=\frac{7}{2}.
   $$

5. **Horizontal Asymptote**

Since the degrees of the numerator and denominator are both $$ 1 $$, the horizontal asymptote is determined by the ratio of the leading coefficients. The leading coefficient of the numerator is $$ 1 $$ and that of the denominator is $$ 2 $$:
   $$
   y=\frac{1}{2}.
   $$
   Thus, the horizontal asymptote is at
   $$
   y=\frac{1}{2}.
   $$

Plotting these features on the graph of 
$$
f(x)=\frac{x+3}{2x-7}
$$
involves marking the intercepts at $$(-3,0)$$ and $$\left(0,-\frac{3}{7}\right)$$, drawing the vertical asymptote as a dashed line at $$ x=\frac{7}{2} $$, and drawing the horizontal asymptote as a dashed line at $$ y=\frac{1}{2} $$. The function will approach these asymptotes but never cross the vertical asymptote, and for large positive or negative $$ x $$, the graph will get arbitrarily close to the horizontal asymptote $$ y=\frac{1}{2} $$.
- **Domain:** All real numbers except $$ x = \frac{7}{2} $$. - **x-intercept:** $$(-3, 0)$$. - **y-intercept:** $$\left(0, -\frac{3}{7}\right)$$. - **Vertical Asymptote:** $$ x = \frac{7}{2} $$. - **Horizontal Asymptote:** $$ y = \frac{1}{2} $$. Plot these on the graph of $$ f(x) = \frac{x+3}{2x-7} $$.

Plot all of the existing five features of the following
If you get a fraction or decimal then plot as close to

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