FINDING DOMAIN AND RANGE OF A FUNCTION IN INTERVAL NOTATION

Find the domain and range range of each of the following functions. Express answers in interval notation.

Problem 1 :

Solution:

For what value of x the function is defined, first we have to check for what values of x the function is undefined.

For that, we have to equate the denominator to 0. 

x + 2 = 0

x = -2

The domain is all real values except -2.

 x ≠ -2

Domain:

(-∞, -2) ∪ (-2, ∞)

Range:

Problem 2 :

Solution:

For what value of x the function is defined, first we have to check for what values of x the function is undefined.

For that, we have to equate the denominator to 0. 

x - 2 = 0

x = 2

The domain is all real values except 2.

 x ≠ 2

Domain:

(-∞, 2) ∪ (2, ∞)

Range:

Problem 3 :

Solution:

For what value of x the function is defined, first we have to check for what values of x the function is undefined.

For that, we have to equate the denominator to 0. 

x - 3 = 0

x = 3

The domain is all real values except 3.

 x ≠ 3

Domain:

(-∞, 3) ∪ (3, ∞)

Range:

Problem 4 :

Solution:

For what value of x the function is defined, first we have to check for what values of x the function is undefined.

For that, we have to equate the denominator to 0. 

x - 3 = 0

x = 3

The domain is all real values except 3.

 x ≠ 3

Domain:

(-∞, 3) ∪ (3, ∞)

Range:

Problem 5 :

g(t) = |2x - 7| + 5

Solution:

g(t) = |2x - 7| + 5

Domain of g(t) is defined for all real values of x.

Domain:

(- ∞, ∞)

Range:

By analyzing the vertex of the absolute value function, we can figure out the range easily.

Vertex is at (7, 5). So, the range is [5, ∞)

Problem 6 :

h(t) = 6- |t - 1| 

Solution:

h(t) = 6 - |t - 1| 

h(t) = - |t - 1| + 6

The absolute value function opens down.

Domain of h(t) is defined for all real values of t.

Domain:

(- ∞, ∞)

Range:

h(t) = - |t - 1| + 6

Vertex is at (1, 6). So, the range is (-∞, 6].

Problem 7 :

Solution:

Domain:

The domain of function f defines by f(x) is the set of all real numbers. 

Domain: (-∞, ∞)

Range:

The range of function is the set of all real numbers.

Range: (-∞, ∞)

Problem 6 :

Solution:

Domain:

Range:

[2, ∞)