QuestionGraph the function $f(x) = \frac{3}{x-2}$ .

Studdy Solution

STEP 1

1. The function $f(x) = \frac{3}{x-2}$ is a rational function.
2. The graph of the function will have vertical and horizontal asymptotes.
3. The domain of the function excludes values that make the denominator zero.

STEP 2

1. Determine the domain of the function.
2. Identify the vertical and horizontal asymptotes.
3. Analyze the behavior of the function near the asymptotes.
4. Plot key points and sketch the graph.

STEP 3

Identify the domain of the function by finding values of $x$ that make the denominator zero.
The denominator is $x - 2$ , which is zero when $x = 2$ .
Therefore, the domain of $f(x)$ is all real numbers except $x = 2$ .

STEP 4

Identify the vertical asymptote by setting the denominator equal to zero.
The vertical asymptote is at $x = 2$ .

STEP 5

Identify the horizontal asymptote by analyzing the behavior of $f(x)$ as $x$ approaches infinity or negative infinity.
Since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is $y = 0$ .

STEP 6

Analyze the behavior of the function near the vertical asymptote $x = 2$ .
As $x \to 2^+$ , $f(x) \to +\infty$ .
As $x \to 2^-$ , $f(x) \to -\infty$ .

STEP 7

Plot key points to help sketch the graph.
For example, calculate $f(x)$ at some values: - $f(0) = \frac{3}{0-2} = -\frac{3}{2}$ - $f(3) = \frac{3}{3-2} = 3$

STEP 8

Sketch the graph using the asymptotes and key points.
Draw the vertical asymptote at $x = 2$ and the horizontal asymptote at $y = 0$ .
Plot the key points $(0, -\frac{3}{2})$ and $(3, 3)$ .
Draw the curve approaching the asymptotes and passing through the key points.
The graph of $f(x) = \frac{3}{x-2}$ is a hyperbola with a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 0$ .

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