Question

Explain why f(x) = x^2+4x+3/x^2-x-2 is not continuous at x = -1.

Answer 1

The value of x = -1 makes the denominator of the function equal to zero. That is why this value is not included in the domain of f(x)

Explanation:

We have the following expression

`f(x) = (x^2+4x+3)/(x^2-x-2)`

Since the division between zero is not defined then the function f(x) can not include the values of x that make the denominator of the function zero.

Now we search that values of x make 0 the denominator factoring the polynomial
`x^2-x-2`

We need two numbers that when adding them get as a result -1 and when multiplying those numbers, obtain -2 as a result.

These numbers are -2 and 1

Then the factors are:

`(x-2) (x + 1)`

We do the same with the numerator

`x^2+4x+3`

We need two numbers that when adding them get as a result 4 and when multiplying those numbers, obtain 3 as a result.

These numbers are 3 and 1

Then the factors are:

`(x+3)(x + 1)`

Therefore

`f(x) = ((x+3)(x+1))/((x-2)(x+1))`

Note that
`((x+1))/((x+1))=1` only if
`x \\eq -1`

So since
`x = -1` is not included in the domain the function has a discontinuity in
`x = -1`