Find the Domain and Range g(x)=x^2+2x+1

Problem

g(x)=x2+2*x+1

Solution

1. Identify the type of function. The function g(x)=x2+2*x+1 is a polynomial (specifically a quadratic function).

2. Determine the domain. Since polynomials are defined for all real numbers, there are no restrictions such as division by zero or square roots of negative numbers.

3. Write the domain in interval notation.

D=(−∞,∞)

4. Rewrite the function in vertex form to find the range. Factor the quadratic expression, which is a perfect square trinomial.

g(x)=(x+1)2

5. Analyze the behavior of the squared term. Since (x+1)2≥0 for all real x the minimum value of the function is 0

6. Determine the range. The function starts at its minimum value of 0 and increases to infinity.

R=[0,∞)

Final Answer

Domain: *(−∞,∞), Range: *[0,∞)

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