Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)
f: [ - 1, 1] → R is given as
Let f(x) = f(y).
∴ f is a one-one function.
It is clear that f: [ - 1, 1] → Range f is onto.
∴ f: [ - 1, 1]→ Range f is one-one and onto and therefore, the inverse of the function:
f: [ - 1, 1] → Range f exists.
Let g: Range f → [ - 1, 1] be the inverse of f.
Let y be an arbitrary element of range f.
Since f: [ - 1, 1] → Range f is onto, we have:
Now, let us define g: Range f → [ - 1, 1] as
∴gof =I[-1, 1]and fog = IRange f
∴ f - 1 = g
⇒
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
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.
Check the injectivity and surjectivity of the following functions:
(i) f : N → N given by f(x) = x2
(ii) f : Z → Z given by f(x) = x2
(iii) f : R → R given by f(x) = x2
(iv) f : N → N given by f(x) = x3
(v) f : Z → Z given by f(x) = x3
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
Consider f : R+ → [– 5, ∞) given by f(x) = 9x2 + 6x – 5. Show that f is invertible
with .
WHERE IS THE RANGE OF THE FUNCTION