1. Domain of the function f(x) = 1/x:
- Interval Notation: (-∞, 0) ∪ (0, ∞)
- Set Builder Notation: {x ∈ ℝ | x ≠ 0}
- Inequalities: x < 0 or x > 0
The domain of the function excludes the value x = 0, where the function is undefined.
2. Vertical asymptote of the function f(x) = 1/x:
The vertical asymptote is at x = 0. As x approaches 0 from the left (negative side), f(x) approaches negative infinity, and as x approaches 0 from the right (positive side), f(x) approaches positive infinity.
3. Coordinates of several points on the graph of f(x) = 1/x:
a. When \( x = -3 \):
\[ f(-3) = 1/(-3) = -1/3 \]
Coordinate: (-3, -1/3)
b. When \( x = -1 \):
\[ f(-1) = 1/(-1) = -1 \]
Coordinate: (-1, -1)
c. When \( y = -2 \):
To find x, set \( f(x) = -2 \):
\[ 1/x = -2 \implies x = -1/2 \]
Coordinate: (-1/2, -2)
d. When \( y = -4 \):
To find x, set \( f(x) = -4 \):
\[ 1/x = -4 \implies x = -1/4 \]
Coordinate: (-1/4, -4)
e. When \( x = 1/4 \):
\[ f(1/4) = 1/(1/4) = 4 \]
Coordinate: (1/4, 4)
f. When \( x = 1/2 \):
\[ f(1/2) = 1/(1/2) = 2 \]
Coordinate: (1/2, 2)
g. When \( y = 1 \):
To find x, set \( f(x) = 1 \):
\[ 1/x = 1 \implies x = 1 \]
Coordinate: (1, 1)
h. When \( y = 1/3 \):
To find x, set \( f(x) = 1/3 \):
\[ 1/x = 1/3 \implies x = 3 \]
Coordinate: (3, 1/3)
4. Description of the graph of the function f(x) = 1/x:
The graph of the function \( f(x) = 1/x \) consists of two branches:
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The left branch (in the second quadrant) approaches the vertical asymptote (x = 0) as x approaches 0 from the left, and it approaches the horizontal asymptote (y = 0) as x approaches negative infinity. It is a downward sloping curve that never touches the axes.
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The right branch (in the first quadrant) approaches the vertical asymptote (x = 0) as x approaches 0 from the right, and it approaches the horizontal asymptote (y = 0) as x approaches positive infinity. It is an upward sloping curve that also never touches the axes.
Both branches are symmetric with respect to the origin, exhibiting odd function properties, which is a characteristic of reciprocal functions.
In summary, the graph reflects the behavior of approaching zero but never reaching it, confirming that \( f(x) = 1/x \) has the characteristics of a hyperbola centered at the origin.