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 .

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**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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