Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Analysis of the Function \( f(x) = 3 \left(\frac{1}{4}\right)^x \)
For the function \( f(x) = 3 \left(\frac{1}{4}\right)^x \), we will explore several key characteristics of its graph. These characteristics include:
- **Growth or Decay**: Determine whether the function represents growth or decay.
- **Domain**: Identify the set of all possible input values (x-values).
- **Range**: Determine the set of all possible output values (y-values).
- **Asymptote**: Identify any lines that the graph approaches but never touches.
- **X-intercept**: Find the point where the graph crosses the x-axis.
- **Y-intercept**: Find the point where the graph crosses the y-axis.
- **End Behavior**: Analyze the behavior of the function as \( x \) approaches \( \infty \) or \( -\infty \).
#### 1. Growth or Decay
Since \( \frac{1}{4} < 1 \), the function \( 3 \left(\frac{1}{4}\right)^x \) represents exponential decay. As \( x \) increases, the value of \( \left(\frac{1}{4}\right)^x \) decreases towards zero.
#### 2. Domain
The domain of \( f(x) = 3 \left(\frac{1}{4}\right)^x \) is all real numbers, written as:
\[ \text{Domain} = (-\infty, \infty) \]
#### 3. Range
As \( x \) increases or decreases, \( f(x) \) will always be positive but never zero. Thus, the range is:
\[ \text{Range} = (0, \infty) \]
#### 4. Asymptote
The graph of \( f(x) = 3 \left(\frac{1}{4}\right)^x \) has a horizontal asymptote at \( y = 0 \). The function approaches this line but never touches it as \( x \) approaches \( \infty \).
#### 5. X-intercept
To find the x-intercept, set \( f(x) \) to 0 and solve for \( x \):
\[ 3 \left(\frac{1}{4}\right)^x =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c9551-2207-4a1f-96cc-82c211723c9b%2F1bb3d891-9d9a-4570-bfc2-712171ad0c2b%2Fbycachb_processed.png&w=3840&q=75)
Transcribed Image Text:### Analysis of the Function \( f(x) = 3 \left(\frac{1}{4}\right)^x \)
For the function \( f(x) = 3 \left(\frac{1}{4}\right)^x \), we will explore several key characteristics of its graph. These characteristics include:
- **Growth or Decay**: Determine whether the function represents growth or decay.
- **Domain**: Identify the set of all possible input values (x-values).
- **Range**: Determine the set of all possible output values (y-values).
- **Asymptote**: Identify any lines that the graph approaches but never touches.
- **X-intercept**: Find the point where the graph crosses the x-axis.
- **Y-intercept**: Find the point where the graph crosses the y-axis.
- **End Behavior**: Analyze the behavior of the function as \( x \) approaches \( \infty \) or \( -\infty \).
#### 1. Growth or Decay
Since \( \frac{1}{4} < 1 \), the function \( 3 \left(\frac{1}{4}\right)^x \) represents exponential decay. As \( x \) increases, the value of \( \left(\frac{1}{4}\right)^x \) decreases towards zero.
#### 2. Domain
The domain of \( f(x) = 3 \left(\frac{1}{4}\right)^x \) is all real numbers, written as:
\[ \text{Domain} = (-\infty, \infty) \]
#### 3. Range
As \( x \) increases or decreases, \( f(x) \) will always be positive but never zero. Thus, the range is:
\[ \text{Range} = (0, \infty) \]
#### 4. Asymptote
The graph of \( f(x) = 3 \left(\frac{1}{4}\right)^x \) has a horizontal asymptote at \( y = 0 \). The function approaches this line but never touches it as \( x \) approaches \( \infty \).
#### 5. X-intercept
To find the x-intercept, set \( f(x) \) to 0 and solve for \( x \):
\[ 3 \left(\frac{1}{4}\right)^x =
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