Question:
Let the function f: R → R be defined by f (x) = cos x, ∀ x ∈ R. Show that f is neither one-one nor onto.
Solution:
We have,
f: R → R, f(x) = cos x
Now,
f (x1) = f (x2)
cos x1 = cos x2
x1 = 2nπ ± x2, n ∈ Z
It’s seen that the above equation has infinite solutions for x1 and x2
Hence, f(x) is many one function.
Also the range of cos x is [-1, 1], which is subset of given co-domain R.
Therefore, the given function is not onto.
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