Question

If f(x) = (x + 2)^2 – 1, what is the largest possible domain of f so that its inverse is also a function? explain by steps

Answer 1

Given info : inverse of a function f (x) = (x + 2)² - 1 is also a function.

To find : largest possible domain of f.

Solution : here, f(x) = (x + 2)² - 1

⇒y = (x + 2)² - 1

⇒±√(y + 1) = x + 2

⇒x = -2 ± √(y + 1)

⇒f(y) = -2 ± √(y + 1) it will be function only when either f(y) = - 2 - √(y + 1) or -2 + √(y + 1) is inverse of f(x).

We know, range of inverse of f(x) = domain of f(x).

so, for large domain, range of inverse of f(x) must be large.

If we choose function f(y) = -2 - √(y + 1)

Domain y ≥ -1

⇒√(y + 1) ≥ 0

⇒-2 - √(y + 1) ≤ - 2

SO range of inverse of f(x) ≤ -2

But if we choose f(y) = -2 + √(y + 1)

range of f(y) ≥ -2

So, for domain of f(x) ≥ -2 , inverse of f is also a function.

Therefore the largest possible domain is [-2, ∞)