m CAT Questions - Algebra: Compound functions

# CAT Practice : Functions

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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.

## Compound functions

Q.4: Consider functions f(x) = x2 + 2x, g(x) = $\sqrt {{\rm{x + 1}}}$ and h(x) = g(f(x)). What are the domain and range of h(x)?

Domain: ( -∞, +∞), Range -[0, ∞]

## Detailed Solution

h(x) = g(f(x)) = g(x2 + 2x) = $\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + 2x + 1}}} {\rm{ = }}\sqrt {{{{\rm{(x + 1)}}}^{\rm{2}}}}$ = |x + 1|

This bit is very important, and often overlooked.

$\sqrt {\rm{9}}$ = 3, not ±3
If x2 = 9, then x can be ±3, but $\sqrt {\rm{9}}$ is only +3.

So, $\sqrt {{{\rm{x}}^{\rm{2}}}}$ = |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, ∞)

Correct Answer: Domain: ( -∞, +∞), Range -[0, ∞]

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Functions.