In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
(i) f: R → R is defined as f(x) = 3 - 4x.
.
∴ f is one-one.
For any real number (y) in R, there existsin R such that
∴f is onto.
Hence, f is bijective.
(ii) f: R → R is defined as
.
.
∴does not imply that x1 = x2
For instance,
∴ f is not one-one.
Consider an element - 2 in co-domain R.
It is seen thatis positive for all x ∈ R.
Thus, there does not exist any x in domain R such that f(x) = - 2.
∴ f is not onto.
Hence, f is neither one-one nor onto.
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.
Let f : N → N be defined by
State whether the function f is bijective. Justify your answer.
Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto,where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R* ?
If f(x) = , show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ?
Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 metres north-west (iii) 40°
(iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2
Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.
Determine order and degree(if defined) of differential equation ym + 2yn + y' =0