# Functions-11

### From Homeworkwiki

**Let f : R → R be a function given by f(x) = x ^{2} + 1. Find**

**(i)f**

^{-1}{-5}**(ii) f**

^{-1}{26}**(iii) f**

^{-1}{10, 37}.**Solution:** Recall that if f : A → B such that y ∈ B. Then f^{-1} {y} = {x ∈ A : f(x) = y}. In other words,

f^{-1} {y} is the set of pre-images of y.

(i) Let f^{-1} {-5} = x. Then f(x) = -5 => x^{2} + 1 = -5

Clearly, this equation is not solvable in R. So, f^{-1} {-5} = Φ.

(ii) Let f^{-1} {26} = x. Then f(x) = 26 => x^{2} + 1 = 26 => x = ± 5.

Therefore, f^{-1} {26} = {-5, 5}.

(iii) Let f^{-1} {10, 37} = x. Then f(x) = 10 or f(x) = 37.

Therefore, x^{2} + 1 = 10 or, x^{2} + 1 = 37 => x = ± 3 or, x = ±6

So, f^{-1} {10, 37} = {3, -3, 6, -6}