To find the domain of the expression [tex]\(\frac{9}{x^2 - 7x}\)[/tex], we need to determine the values of [tex]\(x\)[/tex] for which the denominator [tex]\(x^2 - 7x\)[/tex] is not equal to zero, as division by zero is undefined.
Let's go through the steps:
1. Identify the denominator and set it equal to zero:
[tex]\[
x^2 - 7x = 0
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Factor the equation:
[tex]\[
x(x - 7) = 0
\][/tex]
This gives us two solutions:
[tex]\[
x = 0 \quad \text{or} \quad x = 7
\][/tex]
3. Determine the intervals where the denominator is not zero:
The critical points (where the denominator equals zero) divide the real number line into three intervals. We need to find the intervals on which the denominator [tex]\(x^2 - 7x\)[/tex] is non-zero.
- Interval 1: [tex]\((- \infty, 0)\)[/tex]
- Interval 2: [tex]\((0, 7)\)[/tex]
- Interval 3: [tex]\((7, \infty)\)[/tex]
4. Verify the function's definition over these intervals:
In each of these intervals, the denominator [tex]\(x^2 - 7x\)[/tex] is non-zero, so the function [tex]\(\frac{9}{x^2 - 7x}\)[/tex] is defined.
Thus, the domain of the expression [tex]\(\frac{9}{x^2 - 7x}\)[/tex] is all real numbers except where the denominator is zero. This means the function is defined for all [tex]\(x\)[/tex] except at [tex]\(x = 0\)[/tex] and [tex]\(x = 7\)[/tex].
In interval notation:
[tex]\[
(-\infty, 0) \cup (0, 7) \cup (7, \infty)
\][/tex]
Therefore, the intervals where the function is defined are:
- From negative infinity to 0 (excluding 0),
- From 0 to 7 (excluding 0 and 7),
- From 7 to positive infinity (excluding 7).
