f(x) = ∛x
The given function is f(x) = ∛x, which represents the cubic root of x
The given function is f(x) = ∛x, which represents the cubic root of x. To understand this function, let’s break it down into its components.
1. The symbol “∛” represents the cube root. This means that whatever value is inside the cube root is being raised to the power of 1/3 (or equivalently, divided by 3). For example, ∛8 = 2 because 2^3 = 8.
2. The letter “x” represents the input or the variable of the function. It can be any real number, and when you substitute a value for x, you can find the corresponding output of the function.
Now, let’s analyze the behavior of the function f(x) = ∛x:
1. Domain: The domain of this function is all real numbers because the cube root is defined for any real input.
2. Range: The range of this function is also all real numbers. Since the cube root of any real number will yield a real number.
3. Intercepts: To find the x-intercept, you need to solve the equation f(x) = ∛x = 0. Since a number multiplied by itself three times is zero if and only if the number itself is zero, the only x-intercept is x = 0.
4. Symmetry: This function is not symmetric about the y-axis (x-axis symmetry) or the x-axis (y-axis symmetry) since the cube root function does not exhibit these symmetries.
5. Increasing/Decreasing: The function f(x) = ∛x is increasing for all x because as x increases, the cube root of x also increases. However, it does not increase at a constant rate, as the rate of change depends on the value of x.
6. Local Extrema: This function does not have any local extrema (maximum or minimum) because it continuously increases or decreases.
7. Asymptotes: There are no asymptotes in this function.
8. Graph: The graph of f(x) = ∛x starts at the point (0, 0) and gradually increases or decreases, depending on the sign of x. It will pass through the first quadrant and third quadrant of the coordinate plane.
Remember, this function represents the cubic root of x, and by substituting different values for x, you can find the corresponding output or y-value.
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