Prove that the function f - R - R given by f x -3 x -3 is one-one and onto-

To prove that the function is one one, we have to prove that if f(x_1)=f(x_2), then x_1 = x_2 nbsp;Let f(x_1)=f(x_2) ⇒ 3x_1+3=3x_2+3 ⇒ x_1=x_2 ∴ f(x) is one ...

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Standard XII

Mathematics

Monotonically Increasing Functions

Prove that th...

Question

Prove that the function $f : R \to R$ given by $f (x) = 3 x + 3$ is one-one and onto.

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Solution

To prove that the function is one-one, we have to prove that if $f (x_{1}) = f (x_{2})$ , then $x_{1} = x_{2}$
Let $f (x_{1}) = f (x_{2})$
$\Rightarrow 3 x_{1} + 3 = 3 x_{2} + 3 \Rightarrow x_{1} = x_{2}$
$∴ f (x)$ is one-one.

For onto,
Let $f (x) = y,$ where $y \in R$
$\Rightarrow y = 3 x + 3$
$\Rightarrow x = \frac{y - 3}{3}$
For all $y \in R \Rightarrow x \in R$
$∴ f (x)$ is onto.

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Prove that the function f:R→R given by f(x)=3x+3 is one-one and onto.

To prove that the function is one-one, we have to prove that if f(x1)=f(x2), then x1 = x2 Let f(x1)=f(x2) ⇒3x1+3=3x2+3⇒x1=x2 ∴f(x) is one-one. For onto, Let f(x)=y, where y∈R ⇒y=3x+3 ⇒x=y−33 For all y∈R⇒x∈R ∴f(x) is onto.

Prove that the function $f : R \to R$ given by $f (x) = 3 x + 3$ is one-one and onto.