find-the-range-algrbraically-f-x-x-2-1-

Question Number 70659 by aliesam last updated on 06/Oct/19

f i n d t h e r a n g e a l g r b r a i c a l l y

f (x) = \sqrt{x^{2} - 1}

Answered by MJS last updated on 06/Oct/19

\sqrt{x^{2} - 1} \in R \Rightarrow x^{2} ⩾ 1 \Rightarrow x ⩽ - 1 \lor x ⩾ 1 \Rightarrow

\Rightarrow D = {x \in R ∣ 1 ⩽∣ x ∣}

\frac{d f}{d x} = \frac{x}{\sqrt{x^{2} - 1}} = 0 \Rightarrow no solution for x \in D

- 1 < x \Rightarrow \frac{x}{\sqrt{x^{2} - 1}} < 0 \Rightarrow minimum at the border x = - 1

x > 1 \Rightarrow \frac{x}{\sqrt{x^{2} - 1}} > 0 \Rightarrow minimum at the border x = 1

\Rightarrow range : 0 ⩽ f (x) < + \infty

Facebook Tweet Pin