We are given the function:

$f(x) = \frac{x^2 + x + 1}{x^2 - 6x + 8}$

1. Finding the Domain:

The domain of a rational function is all real numbers except where the denominator equals zero.

Step 1: Set the denominator equal to zero:

$x^2 - 6x + 8 = 0$

Step 2: Factor the quadratic:

$(x - 4)(x - 2) = 0$

Step 3: Solve for $x$:

$x = 4 \quad \text{or} \quad x = 2$

Therefore, the function is undefined at $x = 4$ and $x = 2$. Hence, the domain is all real numbers except $x = 4$ and $x = 2$.

$\text{Domain: } (-\infty, 2) \cup (2, 4) \cup (4, \infty)$

2. Finding the Range:

The range of the function can be more challenging to determine algebraically. However, we can start by analyzing the function's behavior:

• Vertical asymptotes occur at $x = 4$ and $x = 2$, where the function is undefined.
• Horizontal asymptote: Since the degrees of the numerator and denominator are the same (both are quadratic), the horizontal asymptote is determined by the ratio of the leading coefficients. Here, the leading coefficient of both the numerator and denominator is 1, so the horizontal asymptote is:

$y = \frac{1}{1} = 1$

The function will approach $y = 1$ as $x$ tends to $\pm \infty$.

Behavior around asymptotes and specific points:

To fully find the range, we would typically analyze critical points and the behavior of the function near asymptotes. However, based on the structure of the function, we can infer that the range includes all real values except $y = 1$, where the horizontal asymptote lies.

Thus, the range is likely: $\text{Range: } (-\infty, 1) \cup (1, \infty)$

Would you like more detailed steps for range calculation, or any other specific part?

5 Related Questions:

1. How do we find vertical asymptotes for rational functions?
2. What steps are used to determine the horizontal asymptote of a rational function?
3. How can we analyze the critical points of a rational function?
4. What is the importance of factoring polynomials in rational function analysis?
5. How can limits help in finding the behavior of rational functions at infinity?

Tip:

When finding the range, checking the behavior of the function at critical points and limits around vertical asymptotes provides useful insights.