Find the Domain and Range g(x)=(x-3)^2-1

Problem

g(x)=(x−3)2−1

Solution

1. Identify the type of function. The function g(x)=(x−3)2−1 is a quadratic function, which is a polynomial of degree 2.

2. Determine the domain. Since polynomials are defined for all real numbers, there are no restrictions on the input x

D=(−∞,∞)

3. Analyze the vertex form. The function is given in the form g(x)=a*(x−h)2+k where a=1 h=3 and k=−1

4. Find the vertex and direction of the parabola. Since a=1 is positive, the parabola opens upward. The vertex is located at (h,k) which is (3,−1)

5. Determine the range. Because the parabola opens upward, the minimum value of the function is the ycoordinate of the vertex, k=−1 The function takes all values from this minimum to infinity.

R=[−1,∞)

Final Answer

Domain: *(−∞,∞), Range: *[−1,∞)

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