Question: 6. Submit answer Practice similar Determine the domain and range of \( f(x) = -x^2 + 4x + 1 \)…

6. Submit answer Practice similar

Determine the domain and range of \( f(x) = -x^2 + 4x + 1 \). Enter the solution in interval notation.

• Domain: [ ] help (intervals)
• Range: [ ] help (intervals)

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Solution

To find the domain and range of the function \( f(x) = -x^2 + 4x + 1 \), we need to analyze the behavior of the quadratic function. Step 1: Determine the Domain The domain of a polynomial function is all real numbers since there are no restrictions. Domain: \((- \infty, \infty)\) Step 2: Determine the Range This is a quadratic function in the form \( ax^2 + bx + c \) with \( a = -1 \), \( b = 4 \), and \( c = 1 \). Since \( a = -1 \) is negative, the parabola opens downwards, and it has a maximum value at its vertex. Step 3: Find the Vertex The x-coordinate of the vertex is given by \[ x = -\frac{b}{2a} = -\frac{4}{2(-1)} = 2 \] Step 4: Find the Maximum Value (y-coordinate of the Vertex) Substitute \( x = 2 \) into the function to find the maximum value: \[ f(2) = -(2)^2 + 4(2) + 1 \] Calculate step-by-step: \[ f(2) = -4 + 8 + 1 = 5 \] Since the parabola opens downward, this maximum value is the highest point on the graph. Range: \((-\infty, 5]\) Thus, the domain is \((- \infty, \infty)\) and the range is \((-\infty, 5]\).