First slide

Functions (XII)

Question

Which of the following functions is not onto

Moderate

A

$f : R \to R, f (x) = 3 x + 4$
B

$f : R \to R^{+}, f (x) = x^{2} + 2$
C

$f : R^{+} \to R^{+}, f (x) = \sqrt{x}$
D

none of these

Solution

The function $f (x) = 3 x + 4$ is onto as for $y \in$ $R, f (\frac{y - 4}{3}) = y$
The function $f : R^{+} \to R^{+}, f (x) = \sqrt{x}$ is onto as for $y \in R^{+}, f (y^{2}) = y \cdot$
$f : R \to R^{+}, f (x) = x^{2} + 2$ is not onto as $1 \in R^{+}$ has no pre-image.

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