Question

Given a function $f\left(x\right)=\sin x,$ select the correct statements for $f\left(x\right).$

• A. $\frac{\pi }{4}$ is the pre-image of $f\left(x\right)=\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2}$ is the image of $x=\frac{\pi }{6}$
• C. $0^c$ is the pre-image of $f\left(x\right)=1$
• D. $0$ is the image of $x=\frac{\pi }{2}$

Answer 1

Answer: (A), (B)

$\frac{1}{2}$ is the image of $x=\frac{\pi }{6}$

Given: $f\left(x\right)=\sin x$

Now, let's check the validity of statements one by one.

$0^c$ is the pre-image of $f\left(x\right)=1$

This means $f\left(0^c\right)=1,$ which is incorrect as $\sin 0^c=0$

$\frac{\pi }{4}$ is the pre-image of $f\left(x\right)=\frac{1}{\sqrt{2}}$

This means $f(\frac{\pi }{4})=\frac{1}{\sqrt{2}}$
which is true as $\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}$

$\frac{1}{2}$ is the image of $x=\frac{\pi }{6}$

This means $f(\frac{\pi }{6})=\frac{1}{2}$
which is true as $\sin \frac{\pi }{6}=\frac{1}{2}$

$0$ is the image of $x=\frac{\pi }{2}$

This means $f(\frac{\pi }{2})=0$
which is incorrect as $\sin \frac{\pi }{2}=1$