Question
Which statements describe the function \( f(x)=3\left(\frac{1}{3}\right)^{x} \) ? Check all that apply. Each successive output is the previous output divided by 3 . As the domain values increase, the range values decrease. The graph of the function is linear, decreasing from left to right. Each successive output is the previous output multiplied by 3 . The range of the function is all real numbers greater than 0 . The domain of the function is all real numbers greater than 0 .
Ask by Daniels Burton. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
- Each successive output is the previous output divided by 3.
- As the domain values increase, the range values decrease.
- The range of the function is all real numbers greater than 0.
Solution
Let's analyze the function \( f(x) = 3\left(\frac{1}{3}\right)^x \):
1. **Successive Outputs:**
- For each increase in \( x \) by 1, the function becomes:
\[
f(x+1) = 3\left(\frac{1}{3}\right)^{x+1} = 3\left(\frac{1}{3}\right)^x \cdot \frac{1}{3} = \frac{f(x)}{3}
\]
This means each successive output is the previous output **divided by 3**.
2. **Behavior as \( x \) Increases:**
- Since the base \( \frac{1}{3} \) is less than 1, the function decreases as \( x \) increases. Therefore, as the domain values increase, the range values **decrease**.
3. **Graph Characteristics:**
- The function is **exponential**, not linear. It decreases from left to right but is not a straight line.
4. **Range and Domain:**
- **Range:** The function \( f(x) = 3\left(\frac{1}{3}\right)^x \) always yields positive values for any real \( x \). Therefore, the range is **all real numbers greater than 0**.
- **Domain:** The domain of an exponential function includes all real numbers, not just those greater than 0.
Based on this analysis, the correct statements are:
- **Each successive output is the previous output divided by 3.**
- **As the domain values increase, the range values decrease.**
- **The range of the function is all real numbers greater than 0.**
**Correct Options:**
- Each successive output is the previous output divided by 3.
- As the domain values increase, the range values decrease.
- The range of the function is all real numbers greater than 0.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
Let’s break down the function \( f(x) = 3\left(\frac{1}{3}\right)^{x} \) in a fun way! Each successive output is indeed the previous output divided by 3. This function is an exponential decay, so as \( x \) increases, \( f(x) \) steadily decreases! Now, when we look at the graph of this function, it creates a smooth curve that goes down from left to right, not a straight line, confirming that it’s not linear. The range consists of all positive real numbers since the function approaches 0 but never quite touches it! Therefore, the range is greater than 0, while the domain covers all real numbers, so no limits there.
