Question

f: R—>R⁺U{0} f(x)=|x| ,Examine if given function have an inverse. Find inverse, if it exists.

Answer 1

It is given that f: R—>R⁺U{0}, given by f (x) = | x|
We can see that f(-1) = |-1| = 1, f(1) = |1| = 1

⇒ f(-1) = f(1), but -1 ≠ 1.

⇒ f is not one-one.

Now, we consider -1 ϵ R.

We know that f(x) = |x| is always positive

Therefore, there doesn't exist any element x in domain R such that f(x) = |x| = -1
hence, range of f(x) belongs to $\mathbb{R^+}U\{0\}$
co-domain = range
⇒ f is onto.

Therefore, f is onto but not one - one . but we know, any function is inversible only when function is one one as well as onto.
so, f is not inversible function.