To find the domain of the function $f(x) = \frac{2x + 3}{x^2 - 7x - 8}$, we need to identify the values of $x$ for which the function is defined. A rational function is undefined where the denominator equals zero, so we start by setting the denominator equal to zero and solving for $x$.

Step 1: Set the denominator to zero

$x^2 - 7x - 8 = 0$

Step 2: Factor the quadratic expression

We need two numbers that multiply to $-8$ and add to $-7$. These numbers are $-8$ and $1$.

$x^2 - 7x - 8 = (x - 8)(x + 1) = 0$

Step 3: Solve for $x$

Set each factor equal to zero: $x - 8 = 0 \quad \Rightarrow \quad x = 8$ $x + 1 = 0 \quad \Rightarrow \quad x = -1$

So, the function is undefined at $x = 8$ and $x = -1$.

Step 4: State the domain

The domain of the function is all real numbers except where the denominator is zero. Therefore, the domain is:

$\boxed{(-\infty, -1) \cup (-1, 8) \cup (8, \infty)}$

Summary

• The function $f(x) = \frac{2x + 3}{x^2 - 7x - 8}$ is undefined at $x = 8$ and $x = -1$.
• The domain is all real numbers except $x = 8$ and $x = -1$.

Would you like further explanation or have any other questions?

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Here are 5 related questions you might find helpful:

1. What is the range of the function $f(x) = \frac{2x + 3}{x^2 - 7x - 8}$?
2. How do you find vertical asymptotes of rational functions?
3. What are horizontal asymptotes, and does this function have one?
4. How do you factor more complicated quadratic expressions?
5. What is the behavior of the function near its undefined points $x = 8$ and $x = -1$?

Tip: Always check if factors in the denominator also appear in the numerator—this can help identify holes in the graph instead of vertical asymptotes.