Question: Use the graph to complete the following.
(a) Write the equations for all vertical and horizontal asymptotes. Enter the equations using the "and" button as necessary. Select "None" as necessary.
Vertical asymptote(s):
Horizontal asymptote(s):
(b) Find the domain and range of f.
Write each answer as an interval or union of intervals.
Domain:
Range:
(c) Find all x-intercepts and y-intercepts. Check all that apply.
x-intercept(s): -3 1 2 None
y-intercept(s): 1 0 2 None
graph: A rational-function graph with three branches; one branch is in the bottom-left, approaching a horizontal asymptote near $y = -1$ and a vertical asymptote near $x = -3$, another branch in the center crosses near the origin region and falls toward a vertical asymptote near $x = 2$, and the right branch is in the top-right approaching the horizontal asymptote near $y = 1$ and the vertical asymptote near $x = 2$; position_hint = top-left
1. **State the problem:** We analyze the graph of a rational function $f$ to find its vertical and horizontal asymptotes, domain, range, and intercepts.
2. **Vertical asymptotes:** These occur where the function approaches infinity or negative infinity, typically where the denominator is zero.
From the graph description:
- There is a vertical asymptote near $x = -3$.
- There is another vertical asymptote near $x = 2$.
So, vertical asymptotes are at $$x = -3 \text{ and } x = 2.$$
3. **Horizontal asymptotes:** These describe the behavior of $f(x)$ as $x \to \pm \infty$.
From the graph description:
- One branch approaches $y = -1$ as $x \to -\infty$.
- Another branch approaches $y = 1$ as $x \to \infty$.
So, horizontal asymptotes are at $$y = -1 \text{ and } y = 1.$$
4. **Domain:** The domain includes all real numbers except where vertical asymptotes occur.
Thus, domain is $$(-\infty, -3) \cup (-3, 2) \cup (2, \infty).$$
5. **Range:** From the graph description, the function values cover all $y$ except possibly the horizontal asymptotes.
Since the graph approaches but does not cross the horizontal asymptotes, the range excludes $y = -1$ and $y = 1$.
Hence, range is $$(-\infty, -1) \cup (-1, 1) \cup (1, \infty).$$
6. **Intercepts:**
- $x$-intercepts are points where $f(x) = 0$.
From the graph, the center branch crosses near the origin region. The possible $x$-intercepts given are $-3, 1, 2, \text{None}$.
Since $x = -3$ and $x = 2$ are vertical asymptotes, they cannot be $x$-intercepts.
The graph crosses near $x = 1$, so $x$-intercept is $$1.$$
- $y$-intercepts are points where $x=0$.
From the graph, the center branch crosses near the origin region, so the $y$-intercept is near $0$.
Thus, $y$-intercept is $$0.$$