Prove that the function f - -given byf x - 2x- is one-one and onto-

Solve for one one: Given: f : nbsp;ℝ nbsp; → nbsp;ℝ f (x) = 2x f (x_1) = f(x_2) 2x_1= 2x_2 x_1 = x_2 Hence, if f(x_1)= f(x_2), ⇒ x_1= x_2 ∴ function f is ...

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Quantitative Aptitude

A+B+C=N (N Fixed)

Prove that th...

Question

Prove that the function $f : R \to R$ ,given by
$f (x) = 2 x$ , is one-one and onto.

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Solution

Solve for one-one:
Given: $f : R \to R$
$f (x) = 2 x$
$f (x_{1}) = f (x_{2})$
$2 x_{1} = 2 x_{2}$
$x_{1} = x_{2}$
Hence, if $f (x_{1}) = f (x_{2})$ ,
$\Rightarrow x_{1} = x_{2}$
$∴$ function $f$ is one-one.

Solve for onto:
$f : R \to R$
$f (x) = 2 x$
Let $f (x) = y$ such that $y \in R$
$2 x = y$
$x = \frac{y}{2}$
Since $y$ is a real number,
Hence, $\frac{y}{2}$ will also be a real number.
So, $x$ will also be a real number, i.e., $x \in R$
therefore, $f$ is onto.
Hence, $f$ is one-one and onto.

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A+B+C=N (N Fixed)

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Prove that the function f:R→R,given by f(x)=2x, is one-one and onto.

Prove that the function $f : R \to R$ ,given by
$f (x) = 2 x$ , is one-one and onto.