CAT Quantitative Aptitude Questions | CAT Algebra - Functions questions

The question is about Compound functions. We need to find out the domain and range of a function of a function that is called a compound function. Functions is a simple topic which involves a lot of basic ideas. Make sure that you a get of hold of them by solving these questions. A range of CAT questions can be asked based on this simple concept of Functions questions in the CAT Exam. Make sure you practice a lot of questions in CAT Functions Algebra to push your CAT preparation in the right direction. Functions as an idea is a fabulous idea if one can wrap their head around it the right way, and it has been one of the staple ideas tested by the CAT exam historically.

Question 3: 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)?

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Functions: Competitive exams always have questions that have no clear starting point for solving them, but have an easy solution if you think the right way. This is one such question.

h(x) = g(f(x)) = g(x² + 2x) = √(x² + 2x + 1) = √(x + 1)² = |x + 1|
This bit is very important, and often overlooked.

√(9) = 3, not ±3
If x² = 9, then x can be ±3, but √(9) is only +3.

So, √(x²) = |x|, not +x, not ±x
Domain of |x + 1| = ( -∞, +∞), x can take any value.
As far as the range is concerned, |x + 1| cannot be negative. So, range = [0, ∞)

The question is "What are the domain and range of h(x)?"

Hence the answer is "Domain: ( -∞, +∞), Range -[0, ∞]"