Question
Let \( f(x)=\frac{(x+6)}{(x-3)(2 x+5)} \) Find the domain in interval notation Question Help: Use -oo for \( -\infty \), oo for \( \infty \), U for union. Submit Question
Ask by Wells Daniel. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The domain of \( f(x) \) is all real numbers except \( x = 3 \) and \( x = -\frac{5}{2} \). In interval notation, this is:
\[
(-\infty, -\frac{5}{2}) \cup (-\frac{5}{2},\, 3) \cup (3,\, \infty)
\]
Solution
To determine the domain of the function \( f(x) = \frac{x + 6}{(x - 3)(2x + 5)} \), we need to identify all real numbers \( x \) for which the denominator is not zero, since division by zero is undefined.
1. **Set the denominator not equal to zero:**
\[
(x - 3)(2x + 5) \neq 0
\]
2. **Find the values that make the denominator zero:**
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
2x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{2}
\]
3. **Exclude these values from the real numbers:**
\[
x \neq 3 \quad \text{and} \quad x \neq -\frac{5}{2}
\]
4. **Express the domain in interval notation:**
\[
(-\infty, -\frac{5}{2}) \cup (-\frac{5}{2}, 3) \cup (3, \infty)
\]
**Final Answer:**
\[
(-\infty, -\frac{5}{2}) \cup (-\frac{5}{2},\, 3) \cup (3,\, \infty)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To determine the domain of \( f(x) = \frac{(x+6)}{(x-3)(2x+5)} \), we need to find where the function is undefined, which occurs when the denominator equals zero. Setting the denominator to zero gives us the equations \( x - 3 = 0 \) and \( 2x + 5 = 0 \). Solving these, we find \( x = 3 \) and \( x = -\frac{5}{2} \). Hence, the function is undefined at these two points. Therefore, the domain in interval notation is \( (-\infty, -\frac{5}{2}) \cup (-\frac{5}{2}, 3) \cup (3, \infty) \). Remember, we exclude the points where the function is undefined because we cannot divide by zero! Stay vigilant for any other hidden exclusions when dealing with complex functions!
