Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
The function [tex]\( f(x) = 8^{\frac{1}{3} x} \)[/tex] is an exponential function. To understand its behavior and to graph it, let's go through it step by step.
### 1. Understanding the Function
The base of the exponential function is [tex]\( 8^{\frac{1}{3}} \)[/tex]. Since [tex]\( \frac{1}{3} \)[/tex] is the exponent of 8, we can recognize that [tex]\( 8^{\frac{1}{3}} = 2 \)[/tex] because 2 cubed equals 8. Thus, we can rewrite the function as:
[tex]\[ f(x) = 2^x \][/tex]
### 2. Analyzing the Characteristics
- Domain: The function is defined for all real numbers, so the domain is [tex]\( (-\infty, \infty) \)[/tex].
- Range: Since the exponential function only yields positive values, the range is [tex]\( (0, \infty) \)[/tex].
- Y-Intercept: When [tex]\( x = 0 \)[/tex],
[tex]\[ f(0) = 2^0 = 1 \][/tex]
So the y-intercept is at the point [tex]\( (0, 1) \)[/tex].
### 3. Calculating Key Points
To plot the function, let's determine the values of [tex]\( f(x) \)[/tex] for some selected [tex]\( x \)[/tex] values.
| [tex]\( x \)[/tex] | [tex]\( f(x) \)[/tex] |
|---------|--------------|
| -5 | 0.03125 |
| -2.5 | 0.17678 |
| 0 | 1 |
| 2.5 | 5.65685 |
| 5 | 32 |
These points give us a sense of how the function behaves.
### 4. Graphing the Function
To graph the function, we can:
1. Plot the y-intercept at [tex]\( (0, 1) \)[/tex].
2. Note the increasing nature of the exponential function.
3. Identify that as [tex]\( x \)[/tex] becomes more negative, [tex]\( f(x) \)[/tex] approaches zero but never touches the x-axis (hence, an asymptote at [tex]\( y = 0 \)[/tex]).
### 5. Visualizing Data Points
From our calculated values:
- For [tex]\( x = -5 \)[/tex], [tex]\( f(-5) = 0.03125 \)[/tex] - This is very close to 0 but not reaching it.
- For [tex]\( x = -2.5 \)[/tex], [tex]\( f(-2.5) \approx 0.17678 \)[/tex] - Slightly more than 0.
- For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 1 \)[/tex] - At the y-intercept.
- For [tex]\( x = 2.5 \)[/tex], [tex]\( f(2.5) \approx 5.657 \)[/tex] - Higher as [tex]\( x \)[/tex] increases.
- For [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 32 \)[/tex] - Much higher as [tex]\( x \)[/tex] increases further.
### 6. Conclusion
Based on this analysis, we can deduce how the graph of [tex]\( f(x) = 8^{\frac{1}{3}x} \)[/tex] looks:
- It starts very close to the x-axis for large negative [tex]\( x \)[/tex] values.
- Passes through the point [tex]\( (0, 1) \)[/tex] where [tex]\( x = 0 \)[/tex].
- Rises steeply as [tex]\( x \)[/tex] increases due to the exponential nature of the function.
The graph representing [tex]\( f(x) = 8^{\frac{1}{3} x} \)[/tex] will look like a standard exponential growth curve, starting close to zero for negative x-values and increasing rapidly for positive x-values.
### 1. Understanding the Function
The base of the exponential function is [tex]\( 8^{\frac{1}{3}} \)[/tex]. Since [tex]\( \frac{1}{3} \)[/tex] is the exponent of 8, we can recognize that [tex]\( 8^{\frac{1}{3}} = 2 \)[/tex] because 2 cubed equals 8. Thus, we can rewrite the function as:
[tex]\[ f(x) = 2^x \][/tex]
### 2. Analyzing the Characteristics
- Domain: The function is defined for all real numbers, so the domain is [tex]\( (-\infty, \infty) \)[/tex].
- Range: Since the exponential function only yields positive values, the range is [tex]\( (0, \infty) \)[/tex].
- Y-Intercept: When [tex]\( x = 0 \)[/tex],
[tex]\[ f(0) = 2^0 = 1 \][/tex]
So the y-intercept is at the point [tex]\( (0, 1) \)[/tex].
### 3. Calculating Key Points
To plot the function, let's determine the values of [tex]\( f(x) \)[/tex] for some selected [tex]\( x \)[/tex] values.
| [tex]\( x \)[/tex] | [tex]\( f(x) \)[/tex] |
|---------|--------------|
| -5 | 0.03125 |
| -2.5 | 0.17678 |
| 0 | 1 |
| 2.5 | 5.65685 |
| 5 | 32 |
These points give us a sense of how the function behaves.
### 4. Graphing the Function
To graph the function, we can:
1. Plot the y-intercept at [tex]\( (0, 1) \)[/tex].
2. Note the increasing nature of the exponential function.
3. Identify that as [tex]\( x \)[/tex] becomes more negative, [tex]\( f(x) \)[/tex] approaches zero but never touches the x-axis (hence, an asymptote at [tex]\( y = 0 \)[/tex]).
### 5. Visualizing Data Points
From our calculated values:
- For [tex]\( x = -5 \)[/tex], [tex]\( f(-5) = 0.03125 \)[/tex] - This is very close to 0 but not reaching it.
- For [tex]\( x = -2.5 \)[/tex], [tex]\( f(-2.5) \approx 0.17678 \)[/tex] - Slightly more than 0.
- For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 1 \)[/tex] - At the y-intercept.
- For [tex]\( x = 2.5 \)[/tex], [tex]\( f(2.5) \approx 5.657 \)[/tex] - Higher as [tex]\( x \)[/tex] increases.
- For [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 32 \)[/tex] - Much higher as [tex]\( x \)[/tex] increases further.
### 6. Conclusion
Based on this analysis, we can deduce how the graph of [tex]\( f(x) = 8^{\frac{1}{3}x} \)[/tex] looks:
- It starts very close to the x-axis for large negative [tex]\( x \)[/tex] values.
- Passes through the point [tex]\( (0, 1) \)[/tex] where [tex]\( x = 0 \)[/tex].
- Rises steeply as [tex]\( x \)[/tex] increases due to the exponential nature of the function.
The graph representing [tex]\( f(x) = 8^{\frac{1}{3} x} \)[/tex] will look like a standard exponential growth curve, starting close to zero for negative x-values and increasing rapidly for positive x-values.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
