Question:
The domain of definition of the function $f(x) = \frac{\log_3(x+7)}{x^2 - 5x + 6}$ is
The domain of definition of the function $f(x) = \frac{\log_3(x+7)}{x^2 - 5x + 6}$ is
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Always apply domain restrictions separately, then combine them carefully.
Updated On: Apr 30, 2026
- $(-7,\infty)\setminus \{3,2\}$
- $(-3,\infty)\setminus \{3,2\}$
- $(-7,\infty)\setminus \{3\}$
- $(-3,\infty)\setminus \{3\}$
- $(-5,\infty)\setminus \{3\}$
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The Correct Option is A
Solution and Explanation
Concept:
• Log argument $>0$
• Denominator $\neq 0$
Step 1: Log condition.
\[ x+7>0 \Rightarrow x>-7 \]
Step 2: Denominator condition.
\[ x^2-5x+6\neq0 \Rightarrow (x-2)(x-3)\neq0 \] \[ x\neq2, x\neq3 \]
Step 3: Combine.
\[ (-7,\infty)\setminus\{2,3\} \] \[ \boxed{(-7,\infty)\setminus\{2,3\}} \]
• Log argument $>0$
• Denominator $\neq 0$
Step 1: Log condition.
\[ x+7>0 \Rightarrow x>-7 \]
Step 2: Denominator condition.
\[ x^2-5x+6\neq0 \Rightarrow (x-2)(x-3)\neq0 \] \[ x\neq2, x\neq3 \]
Step 3: Combine.
\[ (-7,\infty)\setminus\{2,3\} \] \[ \boxed{(-7,\infty)\setminus\{2,3\}} \]
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