JEE MAIN - Mathematics (2008 - No. 3)
Let $$f:N \to Y$$ be a function defined as f(x) = 4x + 3 where
Y = { y $$ \in $$ N, y = 4x + 3 for some x $$ \in $$ N }.
Show that f is invertible and its inverse is
Y = { y $$ \in $$ N, y = 4x + 3 for some x $$ \in $$ N }.
Show that f is invertible and its inverse is
$$g\left( y \right) = {{3y + 4} \over 4}$$
$$g\left( y \right) = 4 + {{y + 3} \over 4}$$
$$g\left( y \right) = {{y + 3} \over 4}$$
$$g\left( y \right) = {{y - 3} \over 4}$$
Explanation
Clearly $$f$$ is one one and onto, so invertible
Also $$f\left( x \right) = 4x + 3 = y \Rightarrow x = {{y - 3} \over 4}$$
$$\therefore$$ $$\,\,\,\,g\left( y \right) = {{y - 3} \over 4}$$
Also $$f\left( x \right) = 4x + 3 = y \Rightarrow x = {{y - 3} \over 4}$$
$$\therefore$$ $$\,\,\,\,g\left( y \right) = {{y - 3} \over 4}$$
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