The function fx-x2-4 x-30-x2-8 x-18 iA- both one-one and ontoB- many oneC- ontoD- one-one

The correct option is B many onef(x)=x^2+4x+30/x^2 8x+10= (x 2)^2+26/(x 4)^2+2=y clearly f:R → R, f(3)=f(5)=3 for it to be onto ∀ y ϵ R ∃ x ϵ R such that ...

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Standard IX

Mathematics

Many One Function

The function ...

Question

The function $f (x) = \frac{x^{2} + 4 x + 30}{x^{2} - 8 x + 18}$ is

one-one

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onto

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both one-one and onto

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many one

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Solution

The correct option is D many one
$f (x) = \frac{x^{2} + 4 x + 30}{x^{2} - 8 x + 10} = \frac{(x - 2)^{2} + 26}{(x - 4)^{2} + 2} = y$
clearly f:R $\to$ R, $f (3) = f (5) = 3$
for it to be onto $\forall y ϵ R \exists x ϵ R s u c h t h a t f (x) = y$
But here for y = 0 there is no preimage $x$ $ϵ$ R
$∴$ f is not onto

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The function f(x)=x2+4x+30x2−8x+18 is

The correct option is D many onef(x)=x2+4x+30x2−8x+10=(x−2)2+26(x−4)2+2=y clearly f:R → R, f(3)=f(5)=3 for it to be onto ∀ y ϵ R ∃ x ϵ R such that f(x)=y But here for y = 0 there is no preimage x ϵ R ∴ f is not onto

The function $f (x) = \frac{x^{2} + 4 x + 30}{x^{2} - 8 x + 18}$ is