What will be the inverse of the following functions from R to R? (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = x + 1 (c) f: R—>R defined by f(x) = – 3x+4 (d) f: R—>R defined by f(x) = x^3 (e) f: R—>R defined by f(x) = sin (x)

Solution:

(a)

$f(x)=y=x$

Interchange x and y,

$x=y$

Now solving for y, we get,

$\therefore f^{-1}(x)=x$

(b)

$f(x)=y = x + 1$

Interchange x and y,

$x=y+1$

Now solving for y, we get,

$y=x-1 \\\therefore f^{-1}(x)=x-1$

(c)

$f(x)=y = – 3x+4$

Interchange x and y,

$x=-3y+4$

Now solving for y, we get,

$y=\dfrac{x+4}3 \\\therefore f^{-1}(x)=\dfrac{x+4}3$

(d)

$f(x)=y = x^3$

Interchange x and y,

$x=y^3$

Now solving for y, we get,

$y=x^{1/3} \\\therefore f^{-1}(x)=x^{1/3}$

(e)

$f(x)=y = \sin x$

Interchange x and y,

$x=\sin y$

Now solving for y, we get,

$y=\sin^{-1}x \\\therefore f^{-1}(x)=\sin^{-1}x$