# How do you find the domain and range of  ?

Elsie

Solution:

Concept:The domain of a function is the set of all the values of x for which a function  is defined and the value which the function takes is the range of that function.

Also, the domain of a function   is equal to the range of its inverse function   and vice-versa.

For a function having rational expression will have domain of all real number except at the roots of the denominator of the rational function.

Calculation:

Step:1 The function   has denominator  .

The roots of the equation   can be calculated using the quadratic formula as shown below:

Step:2 Hence the roots of the denominator expression are  and  .

Step:3 Therefore, the function has domain as the set of all real numbers except   and 3, which is written in interval notation as  .

Step:4 Now, to find the range of the function, first let   and rearrange the function as quadratic expression.

For the roots to be real in quadratic equation, it must satisfy  .

Substitute the values of a, b and c in the expression  .

Since for all the values of y the inequality satisfies thus the range of the function is set of all real numbers.

Thus, the domain of the function   is   and the range is  .

Answer:The domain of the function   is   and the range is  .

Similar Problems:

Problem 1:What is the domain of a polynomial function.

Solution:

Step:1 The domain of a function is the set of all the values of x for which a function   is defined.

Step:2 For a function having rational expression will have domain of all real number except at the roots of the denominator of the rational function.

Step:3 Since every polynomial function has denominator as a constant value 1 therefore there are no values of x for which the functions is not defined.

Thus, the domain of a polynomial function is the set of all real numbers, which can be written as  .

Answer:The domain of a polynomial function is the set of all real numbers.

Problem 1: What is the domain of a function  .

Solution:

Step:1 The domain of a function is the set of all the values of x for which a function   is defined.

Step:2 For a function having rational expression will have domain of all real number except at the roots of the denominator of the rational function.

Step:3 Since the given function has denominator as a  therefore for the value of  , the functions is not defined.

Step:4 Thus, the domain of a function  is the set of all real numbers except the number 3, which can be written as  .

Answer:The domain of a function   is the set of all real numbers except the number 3, which can be written as  .

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