m CAT Questions - Algebra: Domain and Range

# CAT Practice : Functions

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Finding domain and range of functions is still the most standard type of question from this topic. You should have plenty of practice for these.

## Domain and Range

Q.6: Give the domain and range of the following functions:
1. f(x) = x2 + 1
2. g(x) = log(x + 1)
3. h(x) = 2x
4. f(x) = ${{\rm{1}} \over {{\rm{x + 1}}}}$
5. p(x) = |x + 1|
6. q(x) = [2x], where [x] gives the greatest integer less than or equal to x

## Detailed Solution

1. f(x) = x2 + 1
Domain = All real numbers (x can take any value)
Range [1, ∞). Minimum value of x2 is 0.

2. g(x) = log (x + 1)
Domain = Log of a negative number is not defined so (x + 1) > 0 or x > -1
Domain ( -1, ∞)
Range = (-∞, +∞)

Note: Log is one of those beautiful functions that is defined from a restricted domain to all real numbers. Log 0 is also not defined. Log is defined only for positive numbers

3. h(x) = 2x Domain - All real numbers.
Range = (0, ∞)
The exponent function is the mirror image of the log function.

4. f(x) = ${{\rm{1}} \over {{\rm{x + 1}}}}$
Domain = All real numbers except -1
Range = All real numbers except 0.

5. p(x) = |x + 1|
Domain = All real numbers
Range = [0, ∞) Modulus cannot be negative

6. q(x) = [2x], where [x] gives the greatest integer less than or equal to x
Domain = All real numbers
Range = All integers

7. The range is NOT the set of even numbers. [2x] can be odd. [2 * 0.6] = 1. It is very important to think fractions when you are substituting values.

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Functions.