Find the Domain and Range x^2+4

Problem

ƒ(x)=x2+4

Solution

1. Identify the function type to determine the domain. The expression x2+4 is a polynomial (specifically a quadratic function).

2. Determine the domain by checking for restrictions. Since there are no denominators, square roots of negative numbers, or logarithms, the function is defined for all real numbers.

3. Express the domain in interval notation.

D=(−∞,∞)

4. Identify the vertex to determine the range. The function ƒ(x)=x2+4 is a parabola that opens upward because the coefficient of x2 is positive.

5. Find the minimum value of the function. The smallest value for x2 is 0 (which occurs at x=0.

6. Calculate the y-coordinate of the vertex.

ƒ(0)=0+4=4

7. Determine the range based on the minimum value. Since the parabola opens upward, the function values start at 4 and increase to infinity.

R=[4,∞)

Final Answer

Domain: *(−∞,∞), Range: *[4,∞)

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