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Sagot :
To analyze the given exponential function [tex]\( f(x) = 3\left(\frac{1}{3}\right)^x \)[/tex] and determine which statements are true, we can follow these steps:
1. Determine the initial value of the function:
- The initial value of a function is the value when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = 3\left(\frac{1}{3}\right)^x \)[/tex]:
[tex]\[ f(0) = 3\left(\frac{1}{3}\right)^0 = 3 \cdot 1 = 3. \][/tex]
- Therefore, the initial value is 3, not [tex]\(\frac{1}{3}\)[/tex], so the first option is false.
2. Identify the base of the function:
- The base of the function is the term that is raised to the power of [tex]\(x\)[/tex].
- In the function [tex]\( f(x) = 3\left(\frac{1}{3}\right)^x \)[/tex], the base is [tex]\(\frac{1}{3}\)[/tex].
- Therefore, the second option is true.
3. Determine whether the function shows exponential decay:
- Exponential decay occurs when the base of the exponential function is between 0 and 1.
- Here, the base is [tex]\(\frac{1}{3}\)[/tex], which is between 0 and 1.
- Therefore, the function indeed shows exponential decay, making the third option true.
4. Check if the function is a stretch of [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex]:
- The function [tex]\( f(x) = 3\left(\frac{1}{3}\right)^x \)[/tex] can be thought of as the function [tex]\( \left(\frac{1}{3}\right)^x \)[/tex] multiplied by a constant factor of 3.
- This multiplication by 3 means the function [tex]\( 3\left(\frac{1}{3}\right)^x \)[/tex] is a vertical stretch of the function [tex]\( \left(\frac{1}{3}\right)^x \)[/tex].
- Therefore, the fourth option is true.
5. Check if the function is a shrink of the function [tex]\( f(x) = 3^x \)[/tex]:
- To be a shrink of [tex]\( 3^x \)[/tex], the function would have to be reduced horizontally by some factor, which would mean a base greater than 1.
- In this case, [tex]\( 3\left(\frac{1}{3}\right)^x \)[/tex] does not fit the form of a horizontally shrunk function of [tex]\( 3^x \)[/tex] because it has a different base and exhibits decay, not growth.
- Thus, this option is false.
Concluding, the three correct statements are:
1. The base of the function is [tex]\(\frac{1}{3}\)[/tex].
2. The function shows exponential decay.
3. The function is a stretch of the function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex].
Thus, the correct options to select are:
- The base of the function is [tex]\(\frac{1}{3}\)[/tex].
- The function shows exponential decay.
- The function is a stretch of the function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex].
1. Determine the initial value of the function:
- The initial value of a function is the value when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = 3\left(\frac{1}{3}\right)^x \)[/tex]:
[tex]\[ f(0) = 3\left(\frac{1}{3}\right)^0 = 3 \cdot 1 = 3. \][/tex]
- Therefore, the initial value is 3, not [tex]\(\frac{1}{3}\)[/tex], so the first option is false.
2. Identify the base of the function:
- The base of the function is the term that is raised to the power of [tex]\(x\)[/tex].
- In the function [tex]\( f(x) = 3\left(\frac{1}{3}\right)^x \)[/tex], the base is [tex]\(\frac{1}{3}\)[/tex].
- Therefore, the second option is true.
3. Determine whether the function shows exponential decay:
- Exponential decay occurs when the base of the exponential function is between 0 and 1.
- Here, the base is [tex]\(\frac{1}{3}\)[/tex], which is between 0 and 1.
- Therefore, the function indeed shows exponential decay, making the third option true.
4. Check if the function is a stretch of [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex]:
- The function [tex]\( f(x) = 3\left(\frac{1}{3}\right)^x \)[/tex] can be thought of as the function [tex]\( \left(\frac{1}{3}\right)^x \)[/tex] multiplied by a constant factor of 3.
- This multiplication by 3 means the function [tex]\( 3\left(\frac{1}{3}\right)^x \)[/tex] is a vertical stretch of the function [tex]\( \left(\frac{1}{3}\right)^x \)[/tex].
- Therefore, the fourth option is true.
5. Check if the function is a shrink of the function [tex]\( f(x) = 3^x \)[/tex]:
- To be a shrink of [tex]\( 3^x \)[/tex], the function would have to be reduced horizontally by some factor, which would mean a base greater than 1.
- In this case, [tex]\( 3\left(\frac{1}{3}\right)^x \)[/tex] does not fit the form of a horizontally shrunk function of [tex]\( 3^x \)[/tex] because it has a different base and exhibits decay, not growth.
- Thus, this option is false.
Concluding, the three correct statements are:
1. The base of the function is [tex]\(\frac{1}{3}\)[/tex].
2. The function shows exponential decay.
3. The function is a stretch of the function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex].
Thus, the correct options to select are:
- The base of the function is [tex]\(\frac{1}{3}\)[/tex].
- The function shows exponential decay.
- The function is a stretch of the function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex].
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