Explain, using the theorems, why the function is continuous at every number in its domain.

Question

 

F(x) = 

2x2 − x − 8

x2 + 1

 

a.F(x) is a polynomial, so it is continuous at every number in its domain.

b.F(x) is a rational function, so it is continuous at every number in its domain.

c.F(x) is a composition of functions that are continuous for all real numbers, so it is continuous at every number in its domain.

d.F(x) is not continuous at every number in its domain.

e.none of these

State the domain. (Enter your answer using interval notation.)

Answer 1

Step 1:

The function .

Since the degree of the numerator and the denominator of the function is same, the function is a improper polynomial function.

Domain:

The domain of a function is for all values of x, which makes the function mathematically correct.

Since there shouldn't be any zero in denominator.

The denominator expression is always greater than one.

,for all  values of x.

So the domain of any polynomial function  .

Any polynomial function is continuous on its domain.

Thus the function is continuous at every number on  .

is a rational function, so it is continuous at every number in its domain.

Solution:

Option (b) is correct choice.

is a rational function, so it is continuous at every number in its domain.