Find the Domain and Range f(x)=x^2+2x-3

Problem

ƒ(x)=x2+2*x−3

Solution

1. Identify the type of function. Since ƒ(x)=x2+2*x−3 is a polynomial (specifically a quadratic), 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.

D=(−∞,∞)

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

x=(−2)/(2*(1))=−1

4. Calculate the minimum value of the function by substituting the vertex xcoordinate back into ƒ(x)

ƒ*(−1)=(−1)2+2*(−1)−3

ƒ*(−1)=1−2−3=−4

5. Determine the range. Since the leading coefficient a=1 is positive, the parabola opens upward, meaning the function values start at the minimum y=−4 and go to infinity.

R=[−4,∞)

Final Answer

Domain: *(−∞,∞), Range: *[−4,∞)

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