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Provide step-by-step calculations on how to find the range of a function given by

f(x)=2x ²−4x+3.

User Khaledh
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Final answer:

The range of the function f(x)=2x²−4x+3 is found by determining the vertex, which is the minimum point since the parabola opens upwards. The range is all y-values greater than or equal to 1.

Step-by-step explanation:

To find the range of a function, in this case, f(x)=2x²−4x+3, we need to identify the minimum or maximum value of the function since it is a quadratic function (a parabola). A quadratic function has the form ax2 + bx + c, and its vertex can be found using the vertex formula -b/(2a) for x-coordinate, and then substitute this x-coordinate back into the function to find the y-coordinate (which will be the peak or the trough of the parabola).

Step 1: Use the vertex formula to find the x-coordinate of the vertex. For the function f(x) = 2x2 −4x + 3, a = 2 and b = -4. Thus, the x-coordinate of the vertex is -(-4)/ (2*2) = 4/4 = 1.

Step 2: Substitute this x-coordinate back into the function to find the corresponding y-coordinate, which is f (1) = 2*(1)2 −4*(1) + 3 = 2 - 4 + 3 = 1.

Step 3: Since the coefficient of x2 is positive, the parabola opens upwards, and the vertex is the minimum point of the function. Therefore, the range of the function is y ≥1.

User Shinobii
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