Question

Sa se determine imaginea functiei:
f:(1,4]-->R
f(x)=(3x-1)/(x-1)

Answer 1

$f:(1,4]\rightarrow \mathbb_{R} \\ \\f(x) = \dfrac{3x-1}{x-1} =\dfrac{3x-3+2}{x-1} = \dfrac{3(x-1)+2}{x-1} = \dfrac{3(x-1)}{x-1}+\dfrac{2}{x-1} \Rightarrow \\ \\ \Rightarrow f(x) = 3+\dfrac{2}{x-1}\\ \\ \dfrac{2}{x-1}\rightarrow functie strict descrescatoare pe (1,4] \Rightarrow \\ \\ \Rightarrow f(x) \rightarrow functie strict descrescatoare pe (1,4] \Rightarrow$

$\Rightarrow Imf = \Big[f(4), \lim\limits_{\underset{x\ \textgreater \ 1}{x\rightarrow 1}}}f(x)\Big) \Rightarrow \text{Imf} = \Big[\dfrac{12-1}{4-1},\lim\limits_{\underset{x\ \textgreater \ 1}{x\rightarrow 1}}} \dfrac{3x-1}{x-1}\Big) \Rightarrow \\ \\ \Rightarrow \text{Imf} = \Big[\dfrac{11}{3},\dfrac{2}{0^+}\Big) \Rightarrow \boxed{\text{Imf} = \Big[\dfrac{11}{3},+\infty\Big)}$