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Math  /  Algebra

Questione graph of a rational function ff is shown below. Assume that all asymptotes and intercepts are shown and that the graph has no "holes". 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): x=−1x=-1
Horizontal asymptote(s): □\square (b) Find all xx-intercepts and yy-intercepts. Check all that apply. xx-intercept(s): □\square −1-1 −3-3 □\square - 6 □\square None yy-intercept(s): □\square −6-6 □\square −2-2 □\square −3-3 None (c) Find the domain and range of ff.
Write each answer as an interval or union of intervals. Domain: □\square Range: □\square

Studdy Solution

STEP 1

1. The graph of the rational function f f shows all asymptotes and intercepts.
2. There are no "holes" in the graph.
3. The vertical asymptote is at x=−1 x = -1 .
4. The horizontal asymptote is at y=−2 y = -2 .
5. The x x -intercept is at x=−3 x = -3 .
6. The y y -intercept is at y=−6 y = -6 .

STEP 2

1. Identify and write equations for vertical and horizontal asymptotes.
2. Identify and list all x x -intercepts and y y -intercepts.
3. Determine the domain and range of the function.

STEP 3

The vertical asymptote is given as x=−1 x = -1 .
The horizontal asymptote is given as y=−2 y = -2 .

STEP 4

From the graph, the x x -intercept is at x=−3 x = -3 .
From the graph, the y y -intercept is at y=−6 y = -6 .

STEP 5

The domain of the function is all real numbers except where the vertical asymptote occurs. Therefore, the domain is (−∞,−1)∪(−1,∞) (-\infty, -1) \cup (-1, \infty) .
The range of the function is all real numbers except where the horizontal asymptote occurs. Therefore, the range is (−∞,−2)∪(−2,∞) (-\infty, -2) \cup (-2, \infty) .
The solutions are as follows:
(a) Vertical asymptote(s): x=−1 x = -1
Horizontal asymptote(s): y=−2 y = -2
(b) x x -intercept(s): −3 -3
y y -intercept(s): −6 -6
(c) Domain: (−∞,−1)∪(−1,∞) (-\infty, -1) \cup (-1, \infty)
Range: (−∞,−2)∪(−2,∞) (-\infty, -2) \cup (-2, \infty)

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