Answer: The correct option is
(a) f(n) = n - 1.
Step-by-step explanation: We are given to determine whether the given functions are one-to-one or not.
We know that a function y = f(x) is one-to-one if and only if
f(x) = f(y) ⇔ x = y.
That is, any two distinct elements cannot have the same image.
(a) The given function is
Let us consider that
Similarly,
So, this function is one-to-one.
(b) The given function is
Let us consider that
That is, there may be two unequal elements having same image.
For example, f(-1)=(-1)²+1=1+1=2, f(1)=(1)²+1=1+1=2.
It implies that f(-1)=f(1) but 1 ≠ -1.
So, the given function is not one-to-one.
(c) The given function is
Here, the image of all the elements is 1.
For example,
f(2)=f(3) but 2≠3.
So, more than one element is having the same image and so the function cannot be one-to-one.
(d) The given function is
Here, we see that
So, f(2)=f(3) but 2≠3.
So, the given function is not one-to-one.
Thus, the correct option is (a).