Find the Domain and Range f(x)=x^2+6x-16

Problem

ƒ(x)=x2+6*x−16

Solution

1. Identify the type of function. Since ƒ(x)=x2+6*x−16 is a polynomial (specifically a quadratic), it is defined for all real numbers.

2. Determine the domain. For any polynomial, the input x can be any real value.

3. Find the vertex to determine the range. The xcoordinate of the vertex for a quadratic a*x2+b*x+c is given by x=−b/(2*a)

4. Calculate the xcoordinate of the vertex.

x=−6/(2*(1))

x=−3

5. Evaluate the function at the vertex to find the minimum value, since the parabola opens upward (a>0.

ƒ*(−3)=(−3)2+6*(−3)−16

ƒ*(−3)=9−18−16

ƒ*(−3)=−25

6. Determine the range. Since the minimum value is −25 and the parabola opens upward, the range includes all values greater than or equal to −25

Final Answer

Domain: *(−∞,∞), Range: *[−25,∞)

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