What is the range of the function

Rubi Reid

Rubi Reid

Answered question

2022-03-12

What is the range of the function f(x)=7x24x+4x2+1?

Answer & Explanation

peanut13277tp

peanut13277tp

Beginner2022-03-13Added 2 answers

f(x)=74x+3x2+1=7g(x)
We now have to find the range of g(x)=4x+3x2+1. We can see that the domain of this is (,) and that there are no vertical asymptotes and a horizontal asymptote of y=0.
We can do some more inspection of this function and see that for x34,g(x)>0 and for x<34,g(x)<0. So this means that g(x) looks something like the derivative of a negative normal distribution. Specifically this means that there will be a local maxima at x34 and a local minima at x<34.
g(x)=4(x2+1)8x26x(x2+1)2. Solving for the location of extrema, we get 2x2+3x2=0, which has solutions x={2,0.5}. We already determined which one was the minima and the maxima, so the range of g(x) is [g(2),g(.5)]=[1,4]
So the range of f(x) is [3, 8].

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