Show that the Modulus Function f: R\(\to\)R given by f(x) = IxI, is neither one-one nor onto, where IxI is x, if x is positive or 0 and IxI is −x, if x is negative.
Show that the Modulus Function f: R\(\to\)R given by f(x) = IxI, is neither one-one nor onto, where IxI is x, if x is positive or 0 and IxI is −x, if x is negative.
Solution and Explanation
f: R\(\to\)R is given by,
\[f(x) = |x|= \begin{cases} x, & \quad \text{if } x {\geq0}\\ -x, & \quad \text{if } n {<0} \end{cases}\]It is seen that f(-1)=I-1I=1, f(1)=I1I=1.
∴f(−1) = f(1), but −1 ≠ 1.
∴ f is not one-one.
Now, consider −1 ∈ R.
It is known that f(x) = is always non-negative. Thus, there does not exist any element x in domain R such that f(x) = = −1.
∴ f is not onto.
Hence, the modulus function is neither one-one nor onto.
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Concepts Used:
Types of Functions
Types of Functions
One to One Function
A function is said to be one to one function when f: A → B is One to One if for each element of A there is a distinct element of B.
Many to One Function
A function which maps two or more elements of A to the same element of set B is said to be many to one function. Two or more elements of A have the same image in B.
Onto Function
If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function.
One – One and Onto Function
A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function.
Read More: Types of Functions