PC Guide to Finding Domain of Range of Basic Functions
10/21/2015
Example 1: Find the domain of function f defined by
Solution to Example 1

x can take any real number except 1 since x = 1 would make the
denominator equal to zero and the division by zero is not allowed in
mathematics. Hence the domain in interval notation is given by
(-infinity , 1) U (1 , +infinity)
Matched Problem 1: Find the domain of function f defined by
Example 2: Find the domain of function f defined by
Solution to Example 2

The expression defining function f contains a square root. The
expression under the radical has to satisfy the condition
2x - 8 >= 0 for the function to take real values.

Solve the above linear inequality
x >= 4

The domain, in interval notation, is given by
[4 , +infinity)
Matched Problem 2: Find the domain of function f defined by:
Example 3: Find the domain of function f defined by:
Solution to Example 3

The expression defining function f contains a square root. The
expression under the radical has to satisfy the condition
-x >= 0

Which is equivalent to
x <= 0

The denominator must not be zero, hence x not equal to 3 and x not
equal to -5.

The domain of f is given by
(-infinity , -5) U ( -5 , 0]
Matched Problem 3: Find the domain of function f defined by:
Example 4: Find the range of function f defined by:
Solution to Example 4

The domain of this function is the set of all real numbers. The range is
the set of values that f(x) takes as x varies. If x is a real number, x2 is
either positive or zero. Hence we can write the following:
x 2 >= 0

Subtract -2 to both sides to obtain
x 2 - 2>= -2

The last inequality indicates that x2- 2 takes all values greater than or
equal to -2. The range of f is given by
[ -2 , +infinity)

A graph of f also helps in interpreting the range of a function. Below is
shown the graph of function f given above. Note the lowest point in
the graph has a y (= f (x)) value of -2.
Matched Problem 4: Find the range of function f defined by:
Example 5: Find the Range of function f defined by
Solution to Example 5


f (x) = (x - 1) / (x + 2)
For this rational function, a direct algebraic method similar to those
above is not obvious. Let us first find its inverse, the domain of its
inverse which give the range of f.
We first prove that f is a one to one function and then find its inverse.
For a function to be a one to one, we need to show that
If f(a) = f(b) then a = b.
(a - 1) / (a + 2) = (b - 1) / (b + 2)

Cross multiply, expand and simplify
(a - 1)*(b + 2) = (b - 1)*(a + 2)
a b + 2 a - b - 2 = a b + 2b - a - 2
3a = 3b , which finally gives a = b and proves that f is a one to one
function.

Let us find the inverse of f
y = (x - 1) / (x + 2)

Solve for x
x = (2y + 1) / (1 - y)

change y into x and x into y and write the inverse function
f -1 (x) = y = (2x + 1) / (1 - x)

The range of f is given by the domain of f -1 and is therefore given by
the interval
(-∞ , 1) U (1 , + ∞)
Matched Problem 5: Find the range of function f defined by:
f (x) = ((x - 2)(x + 4)) / ((x - 3) (x + 4))