If f: R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1. Hence, find f−1 (24) and f−1 (5).

Asked by Aaryan | 1 year ago |  214

##### Solution :-

Given f: R → R be defined by f(x) = x3 −3

Now we have to prove that f−1 exists

Injectivity of f:

Let x and y be two elements in domain (R),

Such that, x3 − 3 = y3 − 3

⇒ x3 = y3

⇒ x = y

So, f is one-one.

Surjectivity of f:

Let y be in the co-domain (R)

Such that f(x) = y

⇒ x3 – 3 = y

⇒ x3 = y + 3

⇒ x = $$3\sqrt{(y+3)}$$ in R

⇒ f is onto.

So, f is a bijection and, hence, it is invertible.

Finding f -1:

Let f-1(x) = y…….. (1)

⇒ x= f(y)

⇒ x = y− 3

⇒ x + 3 = y3

⇒ y =$$3\sqrt{(x + 3)}$$ = f-1(x)         [from (1)]

So, f-1(x) = $$3\sqrt{(x + 3)}$$

Now, f-1(24) = $$3\sqrt{(24+3)}$$

=$$3\sqrt{27}$$

$$3\sqrt{3^3}$$

= 3

And f-1(5) =$$3\sqrt{(5 + 3)}$$

$$3\sqrt{8}$$

$$3\sqrt{2^3}$$

= 2

Answered by Sakshi | 1 year ago

### Related Questions

#### Let A be the set of first five natural and let R be a relation on A defined as follows: (x, y) R x ≤ y

Let A be the set of first five natural and let R be a relation on A defined as follows: (x, y) R x ≤ y

Express R and R-1 as sets of ordered pairs. Determine also

(i) the domain of R‑1

(ii) The Range of R.

#### A function f: R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).

A function f: R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).

#### Consider f: R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with f-1(x)

Consider f: R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with f-1(x) = $$\dfrac{\sqrt{(x+6)-1}}{3}$$

If f(x) = $$\dfrac{ (4x + 3)}{(6x – 4)}$$, x ≠ ($$\dfrac{2}{3}$$) show that fof(x) = x, for all x ≠ ($$\dfrac{2}{3}$$). What is the inverse of f?
Consider f: R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1(x) = $$\sqrt{ (x-4)}$$ where R+ is the set of all non-negative real numbers.