Math / Calculus

Question\begin{problem} The graph shown is used to find the following characteristics of the function:
1. The domain and range of the function.
2. The intercepts, if any.
3. Horizontal asymptotes, if any.
4. Vertical asymptotes, if any.
5. Oblique asymptotes, if any.
The domain of the function is $\square$ (Express your answer in interval notation. Use integers or fractions for any numbers in the expression.) \end{problem}

Studdy Solution

STEP 1

1. The function is continuous except at its vertical asymptote.
2. The graph has a vertical asymptote at $x = 0$ .
3. The graph has a horizontal asymptote at $y = 0$ .
4. The graph does not cross its asymptotes.

STEP 2

1. Determine the domain of the function.
2. Determine the range of the function.
3. Identify the intercepts of the function.
4. Identify the horizontal asymptotes.
5. Identify the vertical asymptotes.
6. Identify any oblique asymptotes.

STEP 3

The domain of the function is all real numbers except where the function is undefined.
Since there is a vertical asymptote at $x = 0$ , the function is undefined at $x = 0$ .
Thus, the domain is all real numbers except $x = 0$ .
Domain: $(-\infty, 0) \cup (0, \infty)$

STEP 4

The range of the function is all possible $y$ -values the function can take.
Since there is a horizontal asymptote at $y = 0$ , the function approaches but does not reach $y = 0$ .
The range is all real numbers except $y = 0$ .
Range: $(-\infty, 0) \cup (0, \infty)$

STEP 5

Intercepts are points where the graph crosses the axes.
Since the graph does not cross the $y = 0$ line, there is no $y$ -intercept.
Since the graph does not cross the $x = 0$ line, there is no $x$ -intercept.
Intercepts: None

STEP 6

Horizontal asymptotes are lines that the graph approaches as $x$ approaches infinity or negative infinity.
The graph has a horizontal asymptote at $y = 0$ .
Horizontal Asymptote: $y = 0$

STEP 7

Vertical asymptotes are lines that the graph approaches as $y$ approaches infinity or negative infinity.
The graph has a vertical asymptote at $x = 0$ .
Vertical Asymptote: $x = 0$

STEP 8

Oblique asymptotes occur when the degree of the numerator is one more than the degree of the denominator in a rational function.
There are no oblique asymptotes as the graph does not show any linear behavior as $x$ approaches infinity.
Oblique Asymptotes: None
The domain of the function is:
$\boxed{(-\infty, 0) \cup (0, \infty)}$

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Studdy Solution

STEP 1

1. The function is continuous except at its vertical asymptote.
2. The graph has a vertical asymptote at $x = 0$ .
3. The graph has a horizontal asymptote at $y = 0$ .
4. The graph does not cross its asymptotes.

STEP 2

1. Determine the domain of the function.
2. Determine the range of the function.
3. Identify the intercepts of the function.
4. Identify the horizontal asymptotes.
5. Identify the vertical asymptotes.
6. Identify any oblique asymptotes.

STEP 3

The domain of the function is all real numbers except where the function is undefined.
Since there is a vertical asymptote at $x = 0$ , the function is undefined at $x = 0$ .
Thus, the domain is all real numbers except $x = 0$ .
Domain: $(-\infty, 0) \cup (0, \infty)$

STEP 4

The range of the function is all possible $y$ -values the function can take.
Since there is a horizontal asymptote at $y = 0$ , the function approaches but does not reach $y = 0$ .
The range is all real numbers except $y = 0$ .
Range: $(-\infty, 0) \cup (0, \infty)$

STEP 5

Intercepts are points where the graph crosses the axes.
Since the graph does not cross the $y = 0$ line, there is no $y$ -intercept.
Since the graph does not cross the $x = 0$ line, there is no $x$ -intercept.
Intercepts: None

STEP 6

Horizontal asymptotes are lines that the graph approaches as $x$ approaches infinity or negative infinity.
The graph has a horizontal asymptote at $y = 0$ .
Horizontal Asymptote: $y = 0$

STEP 7

Vertical asymptotes are lines that the graph approaches as $y$ approaches infinity or negative infinity.
The graph has a vertical asymptote at $x = 0$ .
Vertical Asymptote: $x = 0$

STEP 8

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Studdy Solution

STEP 1

1. The function is continuous except at its vertical asymptote.2. The graph has a vertical asymptote at x=0 x = 0 x=0.3. The graph has a horizontal asymptote at y=0 y = 0 y=0.4. The graph does not cross its asymptotes.

STEP 2

1. Determine the domain of the function.2. Determine the range of the function.3. Identify the intercepts of the function.4. Identify the horizontal asymptotes.5. Identify the vertical asymptotes.6. Identify any oblique asymptotes.

STEP 3

STEP 4

STEP 5

Intercepts are points where the graph crosses the axes.Since the graph does not cross the y=0 y = 0 y=0 line, there is no y y y-intercept.Since the graph does not cross the x=0 x = 0 x=0 line, there is no x x x-intercept.Intercepts: None

STEP 6

Horizontal asymptotes are lines that the graph approaches as x x x approaches infinity or negative infinity.The graph has a horizontal asymptote at y=0 y = 0 y=0.Horizontal Asymptote: y=0 y = 0 y=0

STEP 7

Vertical asymptotes are lines that the graph approaches as y y y approaches infinity or negative infinity.The graph has a vertical asymptote at x=0 x = 0 x=0.Vertical Asymptote: x=0 x = 0 x=0

STEP 8

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1. The function is continuous except at its vertical asymptote.
2. The graph has a vertical asymptote at $x = 0$ .
3. The graph has a horizontal asymptote at $y = 0$ .
4. The graph does not cross its asymptotes.

1. Determine the domain of the function.
2. Determine the range of the function.
3. Identify the intercepts of the function.
4. Identify the horizontal asymptotes.
5. Identify the vertical asymptotes.
6. Identify any oblique asymptotes.

Intercepts are points where the graph crosses the axes.
Since the graph does not cross the $y = 0$ line, there is no $y$ -intercept.
Since the graph does not cross the $x = 0$ line, there is no $x$ -intercept.
Intercepts: None

Horizontal asymptotes are lines that the graph approaches as $x$ approaches infinity or negative infinity.
The graph has a horizontal asymptote at $y = 0$ .
Horizontal Asymptote: $y = 0$

Vertical asymptotes are lines that the graph approaches as $y$ approaches infinity or negative infinity.
The graph has a vertical asymptote at $x = 0$ .
Vertical Asymptote: $x = 0$