Alright, let's work through the given function [tex]\( f(x) = \frac{x^2 - 6x + 8}{x + 2} \)[/tex].
1. Domain of the function:
To find the domain of the function, we need to determine where the denominator is zero because division by zero is undefined.
Set the denominator equal to zero:
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
So, the function [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = -2 \)[/tex]. Therefore, the domain of the function in interval notation is:
[tex]\[ (-\infty, -2) \cup (-2, \infty) \][/tex]
2. X-intercepts:
To find the x-intercepts, we need to set the numerator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 6x + 8 = 0 \][/tex]
This is a quadratic equation, and solving it, we get:
[tex]\[ (x-2)(x-4) = 0 \][/tex]
[tex]\[ x = 2 \quad \text{and} \quad x = 4 \][/tex]
Therefore, the x-intercepts are:
[tex]\[ (2, 0) \quad \text{and} \quad (4, 0) \][/tex]
3. Y-intercept:
To find the y-intercept, we evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{0^2 - 6 \cdot 0 + 8}{0 + 2} = \frac{8}{2} = 4 \][/tex]
So, the y-intercept is:
[tex]\[ (0, 4) \][/tex]
Final Answers:
- The domain of the function is:
[tex]\[ (-\infty, -2) \cup (-2, \infty) \][/tex]
- The x-intercepts are:
[tex]\[ (2, 0) \quad \text{and} \quad (4, 0) \][/tex]
- The y-intercept is:
[tex]\[ (0, 4) \][/tex]
