Question

If a function f:R → R is defined by
f(x)=x2- 9x + 22 then f-'(4) =
(A) (-3,6 KB) £3, – 6)
(C) (3,6) (D) none of these​

Answer 1

Solution:

The given function is f(x) = x² - 9x + 22

First we show that f is invertible.

1) Let x₁, x₂ be two distinct elements in the set of real numbers, taken as the domain of f.

f(x₁) = x₁² - 9x₁ + 22, f(x₂) = x₂² - 9x₂ + 22

f(x₁) - f(x₂) = (x₁² - x₂²) - 9 (x₁ - x₂) ≠ 0 [∵ x₁ ≠ x₂]

Since x₁ ≠ x₂ gives f(x₁) ≠ f(x₂), f is injective.

2) Let y be an arbitrary element in the set of real numbers, taken as the co-domain of f.

f(x) = y

or, x² - 9x + 22 = y

or, x² - 9x + (22 - y) = 0

∴ x = {9 ± √(4y - 5)}/2

Since y is a real number, {9 ± √(4y - 7)}/2 is also a real number. Therefore y has pre-image {9 ± √(4y - 7)}/2 in the domain of f. Since y is taken as arbitrary, each element in the co-domain of f has a pre-image under f. Therefore f is surjective.

Since f is injective and surjective, f is a bijection, and hence invertible.

We have found that each element y in the co-domain of f has pre-images {9 ± √(4y - 7)}/2

So f⁻¹ is defined by

f⁻¹(y) ={9 ± √(4y - 7)}/2, y is a real number

or, equivalently f⁻¹(x) = {9 ± √(4x - 7)}/2

Therefore, f⁻¹(4) = (9 ± √9)/2

= (9 ± 3)/2

= 6, 3

= 3, 6

which is the required inverse value.

Thus, option C; (3, 6) is correct.