Question
\( 1 \leftarrow \quad \) Determine the domain of the function. \( f(x)=\frac{x^{2}}{12-x} \)
Ask by Hilton Lyons. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The domain of the function \( f(x)=\frac{x^{2}}{12-x} \) is all real numbers except \( x=12 \).
Solution
1. The function is given by \( f(x)=\frac{x^{2}}{12-x} \).
2. The denominator of the function is \( 12-x \). For the function to be defined, the denominator cannot be zero:
\[
12-x \neq 0
\]
3. Solve the inequality:
\[
12-x \neq 0 \quad \Longrightarrow \quad x \neq 12
\]
4. Since there are no other restrictions (the numerator \( x^2 \) is defined for all \( x \)), the domain of the function is:
\[
\{x \in \mathbb{R} \mid x \neq 12\}
\]
5. In interval notation, the domain is:
\[
(-\infty, 12) \cup (12, \infty)
\]
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Simplify this solution Beyond the Answer
The domain of the function \( f(x) = \frac{x^{2}}{12 - x} \) consists of all real numbers except where the denominator equals zero. To find the restriction, set the denominator equal to zero: \( 12 - x = 0 \). This gives \( x = 12 \). Therefore, the domain is all real numbers except \( x = 12 \), or in interval notation, \( (-\infty, 12) \cup (12, \infty) \). Now, it's important to remember that when dealing with rational functions, always check for values that make the denominator zero, as they are the culprits for undefined behavior! In this case, just avoid \( x = 12 \) and you’re good to go!
