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The range of the function $f(x) = \frac{2}{2+4x^2-4x}$ can be written as an interval $(a,b]$. What is $a+b$?

 Oct 7, 2023
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To find the range of f(x)=2+4x2−4x2​, we can first rewrite it as follows:

f(x) = \frac{2}{2+4(x-1)^2}

Since the denominator is always positive, the function is never undefined. Therefore, the domain of f(x) is all real numbers.

To find the range, we can consider the following cases:

If x<1, then (x−1)2 is positive, so the denominator is greater than 2. Therefore, f(x)<1.

If x=1, then (x−1)2=0, so the denominator is 2. Therefore, f(x)=1.

If x>1, then (x−1)2 is positive, so the denominator is greater than 2. Therefore, f(x)<1.

In conclusion, the range of f(x) is the interval (0,1]​. Therefore, a+b=0+1=1​.

 Oct 7, 2023

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