To find the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts of the rational function [tex]\( r(x) = \frac{5}{x^2 + 6x - 6} \)[/tex], follow these steps:
### Finding the [tex]\( x \)[/tex]-intercepts
The [tex]\( x \)[/tex]-intercepts occur where the function [tex]\( r(x) \)[/tex] equals zero. For a rational function [tex]\( \frac{f(x)}{g(x)} \)[/tex], the numerator [tex]\( f(x) \)[/tex] must be zero for [tex]\( r(x) \)[/tex] to be zero.
The numerator of [tex]\( r(x) \)[/tex] is 5, which is a constant and not equal to zero. This implies that there are no [tex]\( x \)[/tex]-intercepts because [tex]\( 5 \)[/tex] can never be zero. Therefore, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ \text{x-intercept} \quad (x, y) = \text{DNE} \][/tex]
### Finding the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept occurs where the graph of the function intersects the [tex]\( y \)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex]. To find the [tex]\( y \)[/tex]-intercept, evaluate [tex]\( r(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ r(0) = \frac{5}{(0)^2 + 6(0) - 6} = \frac{5}{-6} = -\frac{5}{6} \][/tex]
Therefore, the coordinates of the [tex]\( y \)[/tex]-intercept are:
[tex]\[ \text{y-intercept} \quad (x, y) = (0, -\frac{5}{6}) \][/tex]
### Summary
- [tex]\( x \)[/tex]-intercept: [tex]\((x, y) = \text{DNE}\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((x, y) = (0, -\frac{5}{6})\)[/tex]
