Let's address the questions one by one based on the properties of the reciprocal function \( f(x) = \frac{1}{x} \).
Sketch of the Graph
To visualize the graph of the function \( f(x) = \frac{1}{x} \), it's important to note that the graph has two branches, one in the first quadrant and one in the third quadrant. The graph approaches the x-axis (y = 0) and y-axis (x = 0) but never touches them.
1. Domain of the function
- Interval Notation: The domain excludes \( x = 0 \). Thus, it is \( (-\infty, 0) \cup (0, +\infty) \).
- Set Builder Notation: \( { x \in \mathbb{R} | x \neq 0 } \).
- Inequalities: \( x < 0 \) or \( x > 0 \).
2. Vertical Asymptote
The vertical asymptote occurs where the function is undefined, which is at \( x = 0 \).
Vertical Asymptote: \( x = 0 \).
3. Coordinates of Several Points
To find the y-coordinates and x-coordinates for the given x and y values:
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a. What is the y-coordinate when \( x = -3 \)? \[ f(-3) = \frac{1}{-3} = -\frac{1}{3} \]
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b. What is the y-coordinate when \( x = -1 \)? \[ f(-1) = \frac{1}{-1} = -1 \]
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c. What is the x-coordinate when \( y = -2 \)? To find \( x \) when \( f(x) = -2 \): \[ -2 = \frac{1}{x} \Rightarrow x = \frac{1}{-2} = -\frac{1}{2} \]
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d. What is the x-coordinate when \( y = -4 \)? To find \( x \) when \( f(x) = -4 \): \[ -4 = \frac{1}{x} \Rightarrow x = \frac{1}{-4} = -\frac{1}{4} \]
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e. What is the y-coordinate when \( x = \frac{1}{4} \)? \[ f\left(\frac{1}{4}\right) = 4 \]
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f. What is the y-coordinate when \( x = \frac{1}{2} \)? \[ f\left(\frac{1}{2}\right) = 2 \]
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g. What is the x-coordinate when \( y = 1 \)? To find \( x \) when \( f(x) = 1 \): \[ 1 = \frac{1}{x} \Rightarrow x = 1 \]
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h. What is the x-coordinate when \( y = \frac{1}{3} \)? To find \( x \) when \( f(x) = \frac{1}{3} \): \[ \frac{1}{3} = \frac{1}{x} \Rightarrow x = 3 \]
4. Description of the Graph of the Function \( f(x) = \frac{1}{x} \)
The graph of \( f(x) = \frac{1}{x} \) consists of two branches:
- In the first quadrant, the function approaches the x-axis as \( x \to +\infty \) and approaches the y-axis as \( x \to 0^+ \) (the right side). The values of \( f(x) \) in this region are always positive.
- In the third quadrant, the function approaches the x-axis as \( x \to -\infty \) and approaches the y-axis as \( x \to 0^- \) (the left side). The values of \( f(x) \) in this region are always negative.
The overall shape of the graph is hyperbolic, exhibiting symmetry about the origin (odd function property), and it possesses the characteristics of having vertical and horizontal asymptotes along the y-axis and x-axis, respectively.
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