Question

Find the domain of the function. f(x)= 2x+1/ x^2+x-20 (Enter your answer using interval notation.)

Answer 1

To find the domain of the function \( f(x) = \frac{2x + 1}{x^2 + x - 20} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined wherever the denominator is zero.

First, let's find the values for which the denominator \( x^2 + x - 20 \) is zero. We can factor the quadratic expression:

\[
x^2 + x - 20 = (x - 4)(x + 5)
\]

Now, set the factored form equal to zero:

\[
(x - 4)(x + 5) = 0
\]

This gives us two solutions:

\[
x - 4 = 0 \implies x = 4
\]
\[
x + 5 = 0 \implies x = -5
\]

Thus, the function \( f(x) \) is undefined at \( x = 4 \) and \( x = -5 \).

To find the domain, we need to exclude these values from the set of all real numbers. Therefore, the domain in interval notation is:

\[
(-\infty, -5) \cup (-5, 4) \cup (4, \infty)
\]

In summary, the domain of the function \( f(x) \) is

\[
(-\infty, -5) \cup (-5, 4) \cup (4, \infty).
\]