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I am solving function exercises and i encounter a question. I tried to solve it but i am not sure if it is true or not. Can you help me ?

Question: Find the codomain and the range of the following function , $f:(-4,4) \rightarrow R $ and $f(x)=x^2-6x+4$

My work:

I said that if $-4<x<4$ then $0 \leq x^2 <16$ , $-24<-6x<24$ , so $-20<x^2 -6x +4 <44$. Hence , the codomain of f is equal to $(-20,44)$

To find the range of f , i should find the greatest and lowest value which f has taken , so i placed the end points and critical value of f into the function, for $x=-4 , f(-4)=44$ ,$x=4 , f(4)=-4$ $x=3 , f(3)=-5$ so range of f is equal to $(-5,44)$

Is my solutions correct ? If not , can you help ?

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1 Answer 1

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This is a parabolic function, it is opens upward, need to find its valley at $x=-\frac{b}{2a}$ $= -\frac{-6}{2}$ = 3 , check f(-4) = 44, f(3)= -5, f(4)=-4, so the range is (-5,44).

In your writing, you can not multiply negative values w/o changing the direction of the inequity sign. That is why you are wrong.

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