State whether or not the following functions have a well-defined inverse. If the inverse is well-defined, define it. If it is not well-defined, provide justification. a) f : Z → Z. f(x) = 7x – 7 b) f : R → R. f(x) = 7x – 7 c) A = {a, b, c, d, e}. f : P (A) → {0, 1, 2, 3, 4, 5}. f(x) = |x|. It maps a set to the number of elements it contains.

a) $f: z \rightarrow z \quad f(x)=7 x-7$

Domain is integer, codomain is integer

Take $\quad x=1, \quad \Rightarrow f(x)=0$

 $x=2, \quad \Rightarrow \quad f(x)=7$

Thus, $\quad f(x) \neq 1,2,3,4,5 \ldots$

Range is not same as codomain. It is not onto. Thus, no inverse.

b) $f: R \rightarrow R \quad f(x)=7 x-7$

For $R \rightarrow R \quad f(x)$ is both one-one and onto.

Thus, inverse exist.

$y=7 x-7\\\Rightarrow x=\frac{y+7}{7}\\ \therefore \quad f^{-1}(x)=\frac{x+7}{7}$

c) |x| is not a one-one function. It has no inverse.