To determine the values of [tex]\( x \)[/tex] for which the denominator of the function [tex]\( y = \frac{x+1}{x^2 - 4} \)[/tex] equals zero, follow these steps:
1. Identify the denominator of the function.
The function is given as:
[tex]\[
y = \frac{x + 1}{x^2 - 4}
\][/tex]
The denominator of this function is [tex]\( x^2 - 4 \)[/tex].
2. Set the denominator equal to zero to find the points of discontinuity.
We need to find the values of [tex]\( x \)[/tex] that make the denominator zero:
[tex]\[
x^2 - 4 = 0
\][/tex]
3. Solve the equation [tex]\( x^2 - 4 = 0 \)[/tex].
This is a quadratic equation. To solve it, factor the quadratic expression:
[tex]\[
x^2 - 4 = (x - 2)(x + 2) = 0
\][/tex]
4. Set each factor equal to zero and solve for [tex]\( x \)[/tex].
Solve the equation:
[tex]\[
(x - 2) = 0 \quad \text{or} \quad (x + 2) = 0
\][/tex]
For [tex]\( x - 2 = 0 \)[/tex]:
[tex]\[
x = 2
\][/tex]
For [tex]\( x + 2 = 0 \)[/tex]:
[tex]\[
x = -2
\][/tex]
5. Identify the points of discontinuity.
The points of discontinuity are where the denominator is zero, which we found to be at [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
Therefore, the values of [tex]\( x \)[/tex] that make the denominator equal to zero for the function [tex]\( y = \frac{x+1}{x^2 - 4} \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex]. These are the points of discontinuity.
