To find the \(x\)- and \(y\)-intercepts of the rational function
[tex]\[ r(x) = \frac{6}{x^2 + 5x - 7}, \][/tex]
follow these steps:
### Finding the \(x\)-Intercept
The \(x\)-intercept(s) occur where the function \( r(x) \) equals zero. For a fraction to be zero, the numerator must be zero (assuming the denominator is not zero).
In our case, the numerator of \( r(x) \) is \( 6 \). Setting the numerator equal to zero:
[tex]\[ 6 = 0 \][/tex]
This equation has no solution since \( 6 \) is a non-zero constant. Therefore, there are no \(x\)-intercepts for this function. Thus, we can write:
[tex]\[ x \text{-intercepts: } \][/tex]
Since no solutions exist, it can be represented as:
[tex]\[ (x, y) = \][/tex]
### Finding the \(y\)-Intercept
The \(y\)-intercept occurs where the graph of the function crosses the \(y\)-axis. This happens when \( x = 0 \). To find the \(y\)-intercept, substitute \( x = 0 \) into the function \( r(x) \):
[tex]\[ r(0) = \frac{6}{(0)^2 + 5(0) - 7} \][/tex]
[tex]\[ = \frac{6}{-7} \][/tex]
Therefore, the \( y\)-intercept of the function is:
[tex]\[ (x, y) = (0, -\frac{6}{7}) \][/tex]
### Summary
Putting it all together:
[tex]\[ x \text{-intercepts: } (x, y) = (\text{No intercepts}) \][/tex]
[tex]\[ y \text{-intercepts: } (x, y) = (0, -\frac{6}{7}) \][/tex]
