Find whether the following functions are one-one or many-one and into or onto if $f : D \to R$ where D is a domain.
i) $f (x) = \frac{3 x^{2}}{4 π} - c o s π x$

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Solution

A function is one-to-one if its derivative is always positive or always negative in the given range.
$f (x) = \frac{3 x^{2}}{4 π} - c o s π x f^{'} (x) = \frac{6 x}{4 π} + π s i n x π$
$f (x) = \frac{3 x}{2 π} + π s i n (π x) \to$ This function is +ve for $x > - \frac{2 v θ}{3}$ and it will be –ve for $x < - \frac{25}{3}$
$∴ f^{'} (x)$ is not completely +ve or –ve for all x.
$∴ f^{'} (x) = \frac{3 x^{2}}{4 π} - c o s (π x)$ is many to one.

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Find whether the following functions are one-one or many-one and into or onto if f:D→R where D is a domain. i) f(x)=3x24π−cosπx

Find whether the following functions are one-one or many-one and into or onto if $f : D \to R$ where D is a domain.
i) $f (x) = \frac{3 x^{2}}{4 π} - c o s π x$