Let-f-R-1-be-defined-as-f-x-log-10-3x-2-4x-k-1-10-If-f-x-is-surjective-then-find-the-value-of-k-

Question Number 33818 by rahul 19 last updated on 25/Apr/18

L e t f : R \to [1, \infty) b e d e f i n e d a s

f (x) = \log_{10} (\sqrt{3 x^{2} - 4 x + k + 1} + 10) .

I f f (x) i s s u r j e c t i v e, t h e n f i n d

t h e v a l u e o f k ?

Answered by MJS last updated on 26/Apr/18

this {one}^{'} s easier

\sqrt{3 x^{2} - 4 x + k + 1} must be a real number

and must have exactly one zero

(2 distinct zeros \Rightarrow f (x) not defined for

some x \in R; no zeros \Rightarrow m i n (f (x)) > 1)

3 x^{2} - 4 x + k + 1 = 0

x^{2} - \frac{4}{3} x + \frac{k + 1}{3} = 0

x = \frac{2}{3} \pm \frac{\sqrt{1 - 3 k}}{3}

1 - 3 k = 0 \Rightarrow k = \frac{1}{3}

Commented by rahul 19 last updated on 28/Apr/18

p l s e x p l a i n o n e m o r e t i m e .

I f f (x) h a s e x a c t l y o n e z e r o t h e n

i t w i l l n o t b e d e f i n e d f o r t h a t x i . e

w i l l n o t b e s u j e c t i v e \dots . . ? ?

f o r e g : i f i t b e c o m e s (x - 2)

t h e n f (x) i s n o t d e f i n e d a t x = 2 ?

Commented by MJS last updated on 28/Apr/18

not f (x), I only talk about the root

f (x) = \log_{10} (\sqrt{g (x)} + 10)

g (x) has {\begin{cases} no zero \Rightarrow \sqrt{g (x)} > 0 \land f (x) > 1 \forall x \in R \Rightarrow \\ \Rightarrow \exists y \in [1; \infty [: y \neq f (x) \forall x \in R \\ 1 zero p \Rightarrow \sqrt{g (p)} = 0 \Rightarrow f (p) = 1 \Rightarrow \\ \Rightarrow \forall y \in [1; \infty [: \exists x \in R : y = f (x) \Rightarrow f (x) is surjective \\ 2 zeros p, q \Rightarrow \sqrt{g (x)} \notin R \forall x \in] p; q [\Rightarrow \\ \Rightarrow D = R ∖ x \neq R \forall x \in] p; q [ \end{cases}

Commented by rahul 19 last updated on 29/Apr/18

T h a n k y o u s i r .