f(x) = 1/x
The given function is f(x) = 1/x
The given function is f(x) = 1/x. This is a simple rational function, as it involves a fraction where the numerator is a constant (1) and the denominator is a variable (x).
To understand this function, let’s look at some key aspects:
1. Domain: The domain of the function is the set of all possible values of x for which the function is defined. In this case, since we can divide any non-zero number by another non-zero number, the domain of f(x) = 1/x is all real numbers except x = 0. So, the domain is (-∞, 0) U (0, ∞).
2. Range: The range of the function is the set of all possible values of f(x) for all the x-values in the domain. For our function, as x approaches positive infinity, f(x) approaches 0. Similarly, as x approaches negative infinity, f(x) also approaches 0. Thus, the range of f(x) = 1/x is all real numbers except 0. So, the range is (-∞, 0) U (0, ∞).
3. Symmetry: The given function is not symmetric with respect to the y-axis (vertical axis), which means it does not satisfy the condition f(-x) = f(x) for all x in the domain. However, it does exhibit a type of symmetry called odd symmetry or origin symmetry. This means that the graph of the function is symmetric with respect to the origin (0, 0). In other words, if we reflect any point (x, y) on the graph across the x-axis and y-axis, we get the same function back.
4. Asymptotes: Since the function involves a fraction, there are horizontal and vertical asymptotes associated with it. The vertical asymptote is x = 0, which means as x approaches 0 from either side, the value of f(x) approaches positive or negative infinity. The horizontal asymptote is y = 0, which means as x approaches positive or negative infinity, the value of f(x) approaches 0.
5. Graph: The graph of f(x) = 1/x is a hyperbola. It has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The graph is symmetric about the origin and lies in quadrants I and III. As x approaches positive or negative infinity, the graph approaches the x-axis. The farther away x is from 0, the closer the value of f(x) is to zero. Similarly, as x approaches 0 from either side, the value of f(x) goes to positive or negative infinity.
I hope this detailed explanation helps you understand the given function f(x) = 1/x. If you have any further questions or require more clarification, feel free to ask!
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