Find the Domain and Range f(x)=x^2+3x+1/4

Problem

ƒ(x)=x2+3*x+1/4

Solution

1. Identify the type of function. Since ƒ(x) is a polynomial (specifically a quadratic function), it is defined for all real numbers.

2. Determine the domain. For any polynomial, there are no restrictions such as division by zero or square roots of negative numbers, so the domain is all real numbers.

3. Find the vertex to determine the range. For a quadratic a*x2+b*x+c where a>0 the parabola opens upward, and the minimum value occurs at the vertex x=−b/(2*a)

4. Calculate the xcoordinate of the vertex using a=1 and b=3

x=−3/(2*(1))=−1.5

5. Substitute the xcoordinate back into the function to find the minimum yvalue.

ƒ*(−1.5)=(−1.5)2+3*(−1.5)+0.25

6. Simplify the calculation for the minimum value.

ƒ*(−1.5)=2.25−4.5+0.25=−2

7. Conclude the range. Since the parabola opens upward and the minimum value is −2 the range includes all values greater than or equal to −2

Final Answer

Domain: *(−∞,∞), Range: *[−2,∞)

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