What is the range of the function f(x) = -x^2 + 4?

What is the range of the function f(x) = -x^2 + 4?

Step 1: Identify the function given, which is f(x) = -x^2 + 4.
Step 2: Recognize that this function is a quadratic function because it has an x^2 term.
Step 3: Note that the coefficient of x^2 is negative (-1), which means the parabola opens downwards.
Step 4: Find the vertex of the parabola, which gives the maximum value. The vertex form of a downward-opening parabola is at the highest point.
Step 5: The maximum value of f(x) occurs when x = 0. Calculate f(0): f(0) = -0^2 + 4 = 4.
Step 6: Since the parabola opens downwards, the function can take any value less than or equal to the maximum value of 4.
Step 7: Therefore, the range of the function is all values from negative infinity up to and including 4, which is written as (-∞, 4].

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