+819 The range of the function $f(x) = \frac{2}{2+4x^2-4x}$ can be written as an interval $(a,b]$. What is $a+b$?
+618 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.
