Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

State with reason whether following functions have inverse 

h: {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Show that f: [–1, 1] → R, given by f(x) = `x/(x + 2)`  is one-one. Find the inverse of the function f: [–1, 1] → Range f.

(Hint: For y in Range f, y = `f(x) = x/(x + 2)` for some x in [–1, 1] i.e., `x = (2y)/(1 - y)`)

Consider f: R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Consider f: R₊ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f⁻¹ of given f by `f^(-1)(y) = sqrt(y - 4)`, where R₊ is the set of all non-negative real numbers.

Consider f: R₊ → [–5, ∞) given by f(x) = 9x² + 6x – 5. Show that f is invertible with `f^(-1)(y) = ((sqrt(y + 6) - 1)/3)`.

Let f: X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g₁ and g₂ are two inverses of f. Then for all y ∈ Y, fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f⁻¹ and show that (f⁻¹)⁻¹ = f.

Let f: X → Y be an invertible function. Show that the inverse of f⁻¹ is f, i.e., (f⁻¹)⁻¹ = f.

If f: R → R be given by `f(x) = (3 - x^3)^(1/3)`, then fof(x) is ______.

Let `f: R - {-4/3} → R` be a function defined as `f(x) = (4x)/(3x + 4)`. The inverse of f is map g: Range `f → R - {-4/3}` given by

Let f: W → W be defined as f(n) = n − 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

If f: R → R is defined by f(x) = x2 − 3x + 2, find f(f(x)).

For given vectors,  `veca = 2hati - hatj + 2hatk` and `vecb = -hati  + hatj - hatk`, find the unit vector in the direction of the vector `veca +vecb`.

Integrate the rational function:

`x/((x + 1)(x+ 2))`

Integrate the rational function:

`1/(x^2 - 9)`

Integrate the rational function:

`(3x - 1)/((x - 1)(x - 2)(x - 3))`

Integrate the rational function:

`x/((x-1)(x- 2)(x - 3))`

Integrate the rational function:

`(2x)/(x^2 + 3x + 2)`