To find the range of the function f(x) = x² – 3, we need to analyze how the values of f(x) change as x takes on different real numbers.
First, observe that x² is a quadratic function. The vertex of this parabola opens upwards because the coefficient of x² is positive. The minimum value of x² occurs at x = 0, where:
f(0) = 0² – 3 = -3
This means that the lowest point on the graph of the function occurs at -3. Since x² can take any non-negative value (0 and above), as x varies, the output of f(x) can be determined as follows:
- If x = 0, then f(0) = -3.
- If x is any positive or negative number, the output will be greater than -3.
Thus, as x approaches positive or negative infinity, f(x) will also approach positive infinity.
Based on this analysis, the range of the function f(x) = x² – 3 is all real numbers greater than or equal to -3. In interval notation, we can express the range as:
Range: [-3, ∞)