Example for Identity Function

 An example for identity function can be written in 3 steps:
1. Let $f(x) = x^2$, and $g(x) = \sqrt{x}$.
2. Then $g(f(x)) = g(x^2) = \sqrt{x^2} = x$
• Let us try '3' as the input for $g(f(x))$:
$g(f(3)) = g(3^2) = g(9) = \sqrt{9} = 3$
• Whatever input we give for g(f(x)), the output will be the same. So g(f(x)) is an identity function. This is because, g is the inverse of f and vice versa.
3.Also, $f(g(x)) = f(\sqrt{x}) = \left( \sqrt{x} \right)^2 = x$
• Let us try '25' as the input for $f(g(x))$:
$f(g(25)) = f(\sqrt{25}) = \left( \sqrt{25} \right)^2 = 5^2 = 25$
• Whatever input we give for f(g(x)), the output will be the same. So f(g(x)) is an identity function. This is because, f is the inverse of g and vice versa.

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Let us see an example involving trigonometric functions. It can be written in 3 steps:
1. sin^-1 is the inverse of sin function.
2. Let us use ‘$\frac{1}{2}$’ as the input for $\sin \left(\sin^{-1} (x) \right)$. We get:
$\sin \left(\sin^{-1} \left(\frac{1}{2} \right) \right) = \sin \left(\frac{\pi}{6} \right) = \frac{1}{2}$
• Whatever input we give for $\sin \left(\sin^{-1} (x) \right)$, the output will be the same. So $\sin \left(\sin^{-1} (x) \right)$ is an identity function. This is because, sin^-1 is the inverse of sin and vice versa.
3. Let us use ‘$\frac{\pi}{3}$’ as the input for $\sin^{-1} \left(\sin (x) \right)$. We get:
$\sin^{-1} \left(\sin \left(\frac{\pi}{3} \right) \right) = \sin^{-1} \left(\frac{\sqrt{3}}{2} \right) = \frac{\pi}{3}$
• Whatever input we give for $\sin^{-1} \left(\sin (x) \right)$, the output will be the same. So $\sin^{-1} \left(\sin (x) \right)$ is an identity function. This is because, sin is the inverse of sin^-1 and vice versa.