f(x) = x^2
The function f(x) = x^2 represents a quadratic function
The function f(x) = x^2 represents a quadratic function. In this function, the input value, x, is squared and serves as the exponent to calculate the output value, f(x).
To understand the behavior of this function, let’s examine a few key concepts:
Definition of a Function: A function is a rule or relation that associates each input value (x) with exactly one output value (f(x)). In this case, the function f(x) = x^2 relates each input value x to its corresponding output value f(x), obtained by squaring x.
Graph of the Function: The graph of f(x) = x^2 is a parabola that opens upward. The vertex of the parabola is located at (0,0), which is the lowest point of the graph. As x values move away from 0, the function grows larger. For example, if x = 1, f(x) = 1^2 = 1; if x = 2, f(x) = 2^2 = 4; if x = -1, f(x) = (-1)^2 = 1. Thus, we see that the function squares any real number, resulting in a non-negative output.
Domain and Range: The domain is the set of all possible input values for which the function is defined. In the case of f(x) = x^2, the domain is all real numbers since any real number can be squared. The range, on the other hand, represents the set of all possible output values. For f(x) = x^2, the range is [0, +∞), which means the outputs are all non-negative real numbers or zero.
Symmetry: The graph of f(x) = x^2 is symmetric with respect to the y-axis, meaning that substituting -x for x in the function results in the same value. For example, f(-2) = (-2)^2 = 4, which is the same as f(2), confirming symmetry.
In summary, the equation f(x) = x^2 represents a quadratic function that squares any real number x to calculate the corresponding output, f(x). The graph of this function is a parabola that opens upward and is symmetric with respect to the y-axis. The domain of the function is all real numbers, while the range consists of non-negative real numbers including zero.
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