Ax) = log,(2– x) GRAPH THE FOLLOWING FUNCTION: STATE THE ASYMPTOTE: STATE THE DOMAIN: STATE THE RANGE: STATE THE END BEHAVIOR:

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Graph the Following Function:**
\[ f(x) = \log_3 (2 - x) \]

**State the Asymptote:**

**State the Domain:**

**State the Range:**

**State the End Behavior:**

---

**Explanation of the Function:**

To graph the logarithmic function \( f(x) = \log_3 (2 - x) \), we need to understand its properties.

1. **Asymptote:**
   - The vertical asymptote occurs where the argument of the logarithm equals zero, i.e., \( 2 - x = 0 \).
   - Solving this, we get \( x = 2 \).
   - Therefore, the vertical asymptote is at \( x = 2 \).

2. **Domain:**
   - The domain of the function is determined by the argument of the logarithm being greater than zero.
   - \( 2 - x > 0 \rightarrow x < 2 \).
   - Therefore, the domain of the function is \( (-\infty, 2) \).

3. **Range:**
   - The range of a logarithmic function is all real numbers.
   - Therefore, the range of \( f(x) = \log_3 (2 - x) \) is \( (-\infty, \infty) \).

4. **End Behavior:**
   - As \( x \to 2^- \) (approaching 2 from the left), \( f(x) \to -\infty \).
   - As \( x \to -\infty \), \( f(x) \to \infty \).
Transcribed Image Text:**Graph the Following Function:** \[ f(x) = \log_3 (2 - x) \] **State the Asymptote:** **State the Domain:** **State the Range:** **State the End Behavior:** --- **Explanation of the Function:** To graph the logarithmic function \( f(x) = \log_3 (2 - x) \), we need to understand its properties. 1. **Asymptote:** - The vertical asymptote occurs where the argument of the logarithm equals zero, i.e., \( 2 - x = 0 \). - Solving this, we get \( x = 2 \). - Therefore, the vertical asymptote is at \( x = 2 \). 2. **Domain:** - The domain of the function is determined by the argument of the logarithm being greater than zero. - \( 2 - x > 0 \rightarrow x < 2 \). - Therefore, the domain of the function is \( (-\infty, 2) \). 3. **Range:** - The range of a logarithmic function is all real numbers. - Therefore, the range of \( f(x) = \log_3 (2 - x) \) is \( (-\infty, \infty) \). 4. **End Behavior:** - As \( x \to 2^- \) (approaching 2 from the left), \( f(x) \to -\infty \). - As \( x \to -\infty \), \( f(x) \to \infty \).
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