Question

Which of the following functions is not injective ?

• A. $f:R\to R,f\left(x\right)=2x+7$
• B. $f:[0,\pi ]\to [-1,1],f\left(x\right)=\cos x$
• C. $f:[-\frac{\pi }{2},\frac{\pi }{2}]\to R,f\left(x\right)=2\sin x+3$
• D. $f:R\to [-1,1],f\left(x\right)=\sin x$

Answer 1

The correct option is D $f:R\to [-1,1],f\left(x\right)=\sin x$

A:

$f:R\to R,f\left(x\right)=2x+7$

$\frac{dy}{dx}=2>0\to$ one-one (Injective)

B:

$f:[0,\pi ]\to [-1,1],f\left(x\right)=\cos x$
$\frac{dy}{dx}=\sin x=(+ve)\to$ one-one (Injective)

C:

$f:[-\frac{\pi }{2},\frac{\pi }{2}],f\left(x\right)=2\sin x+3$

$\frac{dy}{dx}=2\cos x=+ve\to$ one-one (Injective)

D:

$f:R\to [-1,1],f\left(x\right)=\sin x$
$\frac{dy}{dx}=\cos x=+ve$ & $-ve\to$ many one

Hence, option D.