Answer :
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.
