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

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

Options:

  1. (-∞, 4]
  2. [0, 4]
  3. [4, ∞)
  4. (-∞, 0)

Correct Answer: (-∞, 4]

Solution: 

The function is a downward-opening parabola with a maximum value of 4, so the range is (-∞, 4].

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

Practice Questions

Q1
What is the range of the function f(x) = -x^2 + 4?
  1. (-∞, 4]
  2. [0, 4]
  3. [4, ∞)
  4. (-∞, 0)

Questions & Step-by-Step Solutions

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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