Answers to questions on Functions posted more than a while ago here

1. Give the domain and range of the following functions

i. f(x) = x^{2} + 1

Domain = All real numbers (x can take any value)

Range [1, infinity). Minimum value of x^{2} is 0.

ii. g(x) = log (x + 1)

Domain = Log of a negative number is not defined so (x + 1) > 0 or x > -1

Domain ( -1, infinity)

Range = (-infinity, +infinity)

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

iii. h(x) = 2

^{x}Domain – All real numbers.Range = (0, infinity)

The exponent function is the mirror image of the log function.

iv. f(x) = 1/ (x+1)

Domain = All real numbers except -1

Range = All real numbers except 0

v. p(x) = | x + 1|

Domain = All real numbers

Domain = All real numbers

Range = [0, infinity) Modulus cannot be negative

vi. q(x) = [2x], where [x] gives the greatest integer less than or equal to x

Domain = All real numbers

Range = All integers

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.

Best wishes for CAT 2012.