f-x-1-log-2x-5-x-1-2-Find-the-domain-of-this-function-

Question Number 204853 by depressiveshrek last updated on 29/Feb/24

f (x) = \sqrt{1 - \log_{(2 x + 5)} ({(x + 1)}^{2})}

Find the domain of this function

Answered by Frix last updated on 29/Feb/24

- \frac{5}{2} < x ⩽ 2 \land x \neq - 2 \land x \neq - 1

We could include the limits

\lim_{x \to - \frac{5}{2}} f (x) = 1

\lim_{x \to - 2} f (x) = \sqrt{2}

\Rightarrow - \frac{5}{2} ⩽ x ⩽ 2 \land x \neq - 1

Answered by TonyCWX08 last updated on 29/Feb/24

S e p a r a t e t h e f u n c t i o n :

\sqrt{1 - {l o g}_{2 x + 5} ({(x + 1)}^{2})}

1

{l o g}_{2 x + 5} ({(x + 1)}^{2})

2 x + 5

{(x + 1)}^{2}

x + 1

D o m a i n \sqrt{1 - {l o g}_{2 x + 5} ({(x + 1)}^{2})}

x \in ⟨ - \frac{5}{2}, - 2 ⟩ \cup ⟨ - 2, 2]

D o m a i n 1

x \in R

D o m a i n {l o g}_{2 x + 5} ({(x + 1)}^{2})

x \in ⟨ - \frac{5}{2}, - 2 ⟩ \cup ⟨ - 2, - 1 ⟩ \cup ⟨ - 1, \infty ⟩

D o m a i n 2 x + 5

x \in R

D o m a i n {(x + 1)}^{2}

x \in R

D o m a i n x + 1

x \in R

F i n d i n g I n t e r s e c t i o n

D o m a i n

x \in ⟨ - \frac{5}{2}, - 2 ⟩ \cup ⟨ - 2, 2]