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Consider f: R \rightarrow R given by f(x)=4 x+3 . Show that f is invertible. Find the inverse of \mathbf{f}.
CBSE | Class 12 | Exercise 2.4 | Function | Maths | RD Sharma
Consider f: R \rightarrow R given by f(x)=4 x+3 . Show that f is invertible. Find the inverse of \mathbf{f}.

Given f: R \rightarrow R given by f(x)=4 x+3

Now we have to show that the given function is invertible.

Consider injection of f:

Let x and y be two elements of domain (R),

Such that f(x)=f(y)

\Rightarrow 4 x+3=4 y+3

\Rightarrow 4 x=4 y

\Rightarrow x=y

So, f is one-one.

Now surjection of f:

Let y be in the co-domain (R),

Such that f(x)=y

\Rightarrow 4 x+3=y

\Rightarrow 4 x=y-3

\Rightarrow x=(y-3) / 4 in R (domain)

\Rightarrow \mathrm{f} is onto.

So, f is a bijection and, hence, it is invertible.

Now we have to find \mathrm{f}^{-1}

Let f^{-1}(x)=y \ldots \ldots (1)

\Rightarrow x=f(y)

\Rightarrow x=4 y+3

\Rightarrow x-3=4 y

\Rightarrow y=(x-3) / 4

Now substituting this value in (1) we get

So, f^{-1}(x)=(x-3) / 4

i

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