Question
what is the domain of the following function: f(x)= x-3/ x^2+6x-27.
Answers
Answered by
GPT-4o mini
To find the domain of the function \( f(x) = \frac{x - 3}{x^2 + 6x - 27} \), we need to determine where the function is defined. The function is undefined wherever the denominator is equal to zero.
Let's find when the denominator \( x^2 + 6x - 27 \) is equal to zero by solving the equation:
\[
x^2 + 6x - 27 = 0
\]
We can factor this quadratic expression. We are looking for two numbers that multiply to \(-27\) and add to \(6\). These numbers are \(9\) and \(-3\). Thus, we can factor the expression as:
\[
(x + 9)(x - 3) = 0
\]
Setting each factor equal to zero gives us the solutions:
1. \( x + 9 = 0 \) \(\Rightarrow x = -9\)
2. \( x - 3 = 0 \) \(\Rightarrow x = 3\)
Thus, the function \( f(x) \) is undefined at \( x = -9 \) and \( x = 3 \).
Therefore, the domain of the function is all real numbers except these two points. In interval notation, the domain can be expressed as:
\[
(-\infty, -9) \cup (-9, 3) \cup (3, \infty)
\]
Let's find when the denominator \( x^2 + 6x - 27 \) is equal to zero by solving the equation:
\[
x^2 + 6x - 27 = 0
\]
We can factor this quadratic expression. We are looking for two numbers that multiply to \(-27\) and add to \(6\). These numbers are \(9\) and \(-3\). Thus, we can factor the expression as:
\[
(x + 9)(x - 3) = 0
\]
Setting each factor equal to zero gives us the solutions:
1. \( x + 9 = 0 \) \(\Rightarrow x = -9\)
2. \( x - 3 = 0 \) \(\Rightarrow x = 3\)
Thus, the function \( f(x) \) is undefined at \( x = -9 \) and \( x = 3 \).
Therefore, the domain of the function is all real numbers except these two points. In interval notation, the domain can be expressed as:
\[
(-\infty, -9) \cup (-9, 3) \cup (3, \infty)
\]