Question

Determine all numbers at which the function is continuous.

Answer 1

Option C. is the correct option.

Explanation:

The given function is
`f(x)=(x^(2)-7x+10)/(x^(2)-14x+45)`

Now we will rewrite the given function in the factorial form

`f(x)=(x^(2)-5x-2x+10)/(x^(2)-9x-5x+45)`

`f(x)=((x-2)(x-5))/((x-5)(x-9))`

Now the given function is defined if the denominator is not equal to zero

Therefore for (x-5) = 0

x = 5

and x - 9 =0

x = 9

The function is not defined.

Therefore option C. continuous at every real number except x = 5 and x = 9 is the correct answer.

Answer 2

Option C.

Explanation:

The function is discontinuous at points where the denominator is equal to zero. Then we must take the polynomial that is in the denominator of the expression and look for what points is equal to zero.

Then we have the following expression:

`x ^ 2 -14x +45 = 0`

We must factor it.

Then we look for 2 numbers that multiplied give as a result 45, and added as a result -14.

You can verify that these numbers are -9 and -5.

Then the polynomial is like:

`(x-9)(x-5) = 0`

Then
`x ^ 2 -14x +45 = 0` when
`x = 9` and when
`x = 5`

Therefore the function is continuous in all reals number except for
`x = 9` and in
`x = 5`

The correct option is:

Option c