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

Problem

ƒ(x)=(x−2)2+2

Solution

1. Identify the type of function. The given function ƒ(x)=(x−2)2+2 is a quadratic function written in vertex form ƒ(x)=a*(x−h)2+k

2. Determine the vertex of the parabola. By comparing the given function to the vertex form, we find a=1 h=2 and k=2 The vertex is located at (2,2)

3. Analyze the direction of the parabola. Since the leading coefficient a=1 is positive (a>0, the parabola opens upward.

4. Establish the minimum value. Because the parabola opens upward, the ycoordinate of the vertex represents the minimum value of the function. The minimum value is y=2

5. Determine the range. Since the function has a minimum value of 2 and extends infinitely upward, the range consists of all real numbers greater than or equal to 2

Final Answer

Range:[2,∞)

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