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 ?


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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

	$p l s e x p l a i n o n e m o r e t i m e .$ $I f f (x) h a s e x a c t l y o n e z e r o t h e n$ $i t w i l l n o t b e d e f i n e d f o r t h a t x i . e$ $w i l l n o t b e s u j e c t i v e . . . . . ? ?$ $f o r e g : i f i t b e c o m e s (x - 2)$ $t h e n f (x) i s n o t d e f i n e d a t x = 2 ?$
	Commented by MJS last updated on 28/Apr/18
	$not f(x), I only talk about the root f(x)=log_(10) ((√(g(x)))+10) g(x) has { ((no zero ⇒ (√(g(x)))>0 ∧ f(x)>1 ∀x∈R ⇒)),((⇒ ∃y∈[1; ∞[: y≠f(x)∀x∈R)),((1 zero p ⇒ (√(g(p)))=0 ⇒ f(p)=1 ⇒)),((⇒ ∀y∈[1; ∞[: ∃x∈R: y=f(x) ⇒ f(x) is surjective)),((2 zeros p, q ⇒ (√(g(x)))∉R ∀x∈]p; q[ ⇒)),((⇒ D=R\x≠R ∀x∈]p; q[)) :}$
	$not f (x), I only talk about the root$ $f (x) = \log_{10} (\sqrt{g (x)} + 10)$ $g (x) has {\begin{cases} no zero \Rightarrow \sqrt{g (x)} > 0 \land f (x) > 1 \forall x \in R \Rightarrow \\ \Rightarrow \exists y \in [1; \infty [: y \neq f (x) \forall x \in R \\ 1 zero p \Rightarrow \sqrt{g (p)} = 0 \Rightarrow f (p) = 1 \Rightarrow \\ \Rightarrow \forall y \in [1; \infty [: \exists x \in R : y = f (x) \Rightarrow f (x) is surjective \\ 2 zeros p, q \Rightarrow \sqrt{g (x)} \notin R \forall x \in] p; q [\Rightarrow \\ \Rightarrow D = R ∖ x \neq R \forall x \in] p; q [ \end{cases}$
	Commented by rahul 19 last updated on 29/Apr/18

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