CAT Functions

Questions from functions have increased over the past few years. Given below is an interesting question from functions.

Question

Consider functions  f(x) = x² + 2x,  g(x) = √(x +1) and h(x) = g(f(x)). What are the domain and range of h(x)?

Correct Answers:
Domain = ( - infinity, +infinity)
Range - [0, infinity] 

Explanatory Answer

h(x) = g(f(x)) = g(x² + 2x) = Ö( x² + 2x + 1) = Ö(x+1)² = | x + 1|

This bit is very important, and often overlooked

sqrt(9) = 3, not +3

If x² = 9, then x can be +3, but Sqrt(9) is only +3.

So, Sqrt(x²)= |x|, not +x, not + x

Domain of | x + 1| = ( -infinity, + infinity), x can take any value.

As far as the range is concerned, | x + 1| cannot be negative. So, range = [0, infinity)