Question

When graphing the rational function f(x)=x2+2x−3x+3
, where is the point of discontinuity located?

Select the correct answers from the lists.

(1 point)
The point of discontinuity is located at
because the factor
is in the numerator and denominator.

Answer 1

To identify any points of discontinuity in the rational function

\[ f(x) = \frac{x^2 + 2x - 3}{x + 3}, \]

we start by factoring the numerator.

The numerator \(x^2 + 2x - 3\) can be factored as follows:

1. We need to find two numbers that multiply to \(-3\) and add to \(2\). The numbers \(3\) and \(-1\) fit this requirement.
2. Thus, we have:

\[ x^2 + 2x - 3 = (x + 3)(x - 1). \]

Now, substituting back into the function gives us:

\[ f(x) = \frac{(x + 3)(x - 1)}{x + 3}. \]

Here, we can see that the term \(x + 3\) appears in both the numerator and the denominator. Therefore, we can simplify the function (for \(x \neq -3\)) to:

\[ f(x) = x - 1, \quad \text{for } x \neq -3. \]

However, there is a point of discontinuity at \(x = -3\) because the function is undefined there (the denominator equals zero).

Thus, we can fill in the blanks:

The point of discontinuity is located at **-3** because the factor **x + 3** is in the numerator and denominator.