Exploring the Key Aspects of the Function f(x) = x^2: Domain, Range, Axis of Symmetry, Vertex, Graph, Increasing and Decreasing Intervals, and Axis Intercepts

f(x)=x^2

f(x) = x^2, we can discuss several key aspects

f(x) = x^2, we can discuss several key aspects.

1. Domain:
The domain of a function represents all possible input values. For the function f(x) = x^2, the domain is all real numbers. Any real number can be squared to obtain a real number as the output.

2. Range:
The range of a function represents all possible output values. For the function f(x) = x^2, the range is the set of non-negative real numbers, including zero. The function will always output a positive value or zero when the input is zero, and it will produce increasingly larger positive values as the input increases.

3. Axis of Symmetry:
The axis of symmetry is a vertical line that divides the parabola symmetrically. In the case of f(x) = x^2, the axis of symmetry is the vertical line passing through the vertex of the parabola. Since the equation does not include any coefficients or other transformations, the axis of symmetry is the y-axis, which corresponds to x = 0.

4. Vertex:
The vertex is the lowest or highest point on the parabola. For f(x) = x^2, the vertex is at the point (0, 0), since the coefficient of x^2 is positive. The vertex represents the minimum point of the parabola, and for this function, it lies at the origin.

5. Graph:
The graph of f(x) = x^2 is a symmetric U-shaped curve called a parabola. When graphed on a Cartesian plane, it opens upward and its vertex lies at the origin (0, 0). As x increases, the function value increases quadratically, and as x decreases, the function value also increases quadratically. The graph of f(x) = x^2 is always positive or zero.

6. Increasing and Decreasing Intervals:
The function f(x) = x^2 is increasing for x > 0, meaning that as x increases, the function value also increases. It is decreasing for x < 0, as decreasing x results in increasing negative values. At x = 0, the function is neither increasing nor decreasing. 7. Axis Intercepts: To find the x-intercepts (also known as roots), we set f(x) = x^2 to zero and solve for x. Thus, x^2 = 0, which only occurs when x = 0. Therefore, the parabola intersects the x-axis at the point (0, 0). The y-intercept occurs when x = 0. Substituting x = 0 into f(x) = x^2 gives f(0) = 0^2 = 0. Hence, the parabola intersects the y-axis at the point (0, 0). These are the main components and characteristics of the function f(x) = x^2.

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