To determine whether the given function [tex]\( y = \frac{x}{x^2 - 4} \)[/tex] has an [tex]\( x \)[/tex]-intercept that is equal to its [tex]\( y \)[/tex]-intercept, we need to find both intercepts step-by-step and compare them.
### Finding the [tex]\( x \)[/tex]-intercept:
The [tex]\( x \)[/tex]-intercept occurs where [tex]\( y = 0 \)[/tex]. Therefore, we set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{x}{x^2 - 4} = 0
\][/tex]
A fraction is equal to zero when its numerator is zero (provided that the denominator is not zero). Thus, we solve:
[tex]\[
x = 0
\][/tex]
For [tex]\( x = 0 \)[/tex], the denominator is [tex]\( x^2 - 4 = 0^2 - 4 = -4 \)[/tex], which is not zero. Thus, the equation [tex]\( x = 0 \)[/tex] is valid. Therefore, the [tex]\( x \)[/tex]-intercept is:
[tex]\[
x = 0
\][/tex]
### Finding the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. Therefore, we substitute [tex]\( x = 0 \)[/tex] into the function and solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{0}{0^2 - 4} = \frac{0}{-4} = 0
\][/tex]
Hence, the [tex]\( y \)[/tex]-intercept is:
[tex]\[
y = 0
\][/tex]
### Comparison:
Both the [tex]\( x \)[/tex]-intercept and the [tex]\( y \)[/tex]-intercept are [tex]\( 0 \)[/tex]. Therefore, the [tex]\( x \)[/tex]-intercept is equal to the [tex]\( y \)[/tex]-intercept.
### Conclusion:
The given function [tex]\( y = \frac{x}{x^2 - 4} \)[/tex] indeed has an [tex]\( x \)[/tex]-intercept that is equal to its [tex]\( y \)[/tex]-intercept, and both intercepts are at the point [tex]\( (0, 0) \)[/tex].
