Analyzing the Function f(x) = 1/x: Graph, Domain, Range, and Key Features

f(x)=(1/x)

To analyze the function f(x) = 1/x, we can consider its graph, domain, range, and key features

To analyze the function f(x) = 1/x, we can consider its graph, domain, range, and key features.

1. Graph: The graph of f(x) = 1/x is a hyperbola. It approaches but never reaches the x-axis and the y-axis, forming two branches.

2. Domain: The domain of the function consists of all real numbers except x = 0, as division by zero is undefined.

3. Range: The range of the function also consists of all real numbers except y = 0. As x gets very large (positive or negative), f(x) approaches zero. When x is positive, f(x) is positive, and when x is negative, f(x) is negative.

4. Key Features:
a. x-intercept: To find the x-intercept, we set f(x) = 0: 1/x = 0. Since this equation has no real solutions (division by zero), there is no x-intercept.
b. y-intercept: To find the y-intercept, we set x = 0: f(0) = 1/0. However, this is undefined, so there is no y-intercept.
c. Asymptotes: The graph approaches the x-axis and the y-axis but never touches them.
– Vertical asymptote: The graph becomes infinitely large when x approaches 0 from the positive or negative side.
– Horizontal asymptote: The graph becomes infinitely close to the x-axis (y = 0) as x approaches positive or negative infinity.

5. Symmetry: The function f(x) = 1/x is neither even nor odd. It does not exhibit any symmetry.

6. Increasing/Decreasing: The function is decreasing for x > 0 and increasing for x < 0. It has a global minimum at x = 0 (which is undefined). Overall, the function f(x) = 1/x is a hyperbola that is defined for all real numbers except x = 0. It approaches but never reaches the x-axis or the y-axis, and has vertical asymptote x = 0 and horizontal asymptote y = 0.

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