The range of a cubic function, which is a polynomial function of degree 3, depends on the specific coefficients of the function. However, in general, the range of a cubic function can be either:
1. All real numbers: This is the case for the basic cubic function f(x) = x^3, where the graph of the function passes through every point on the y-axis. The leading coefficient (x^3) is positive, which means the function increases without bound as x approaches positive or negative infinity.
2. A specific interval of real numbers: This is the case for cubic functions with different leading coefficients or non-zero constant terms. The specific interval depends on the analysis of the function's critical points and end behavior.
Determining the Range:
To determine the range of a specific cubic function, one can follow these steps:
Analyze the leading coefficient: If the leading coefficient (the coefficient of the x^3 term) is positive, the function increases without bound as x approaches positive and negative infinity. This suggests a range of all real numbers.
Identify critical points: Find the points where the derivative of the function is zero or undefined. These points divide the number line into intervals where the function is increasing or decreasing.
Evaluate at critical points: Evaluate the function at each critical point and at the endpoints of the number line. This helps determine the maximum and minimum values of the function.
Determine the range: Based on the analysis of critical points and end behavior, determine the specific interval that represents the range of the function.
It's important to note that the range of a cubic function is always connected, meaning it consists of a single interval or a union of two or more connected intervals.
Examples:
f(x) = x^3: Range: All real numbers (-∞, ∞)
f(x) = -x^3: Range: All real numbers (-∞, ∞)
f(x) = x^3 + 2x^2 - 4x + 1: Range: (-1, ∞)
Remember that the range of a cubic function can vary depending on the specific function's coefficients. Analyzing critical points and end behavior is crucial for determining the exact range.