Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R,$ defined by $f(x) = \sin^2x + \cos^2x$

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Answer

$f : R \rightarrow R,$ defined by $f(x) = \sin^2x + \cos^2x$
$f(x) = \sin^2x + \cos^2x = 1$
So, $f(x) = 1$ for every $x$ in $R.$
So, for all elements in the domain, the image is $1.$
So, $f$ is not an injection.
Range of $f = \{1\}$
Co-domain of $f = R$
Both are not same.
So, $f$ is not a surjection and $f$ is not a bijection.

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