Answer :
To analyze the given exponential function [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] and determine the correct statements, we can break the problem into parts and carefully examine each aspect.
1. Initial Value of the Function:
- The initial value of [tex]\( f(x) \)[/tex] is found by evaluating the function at [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3 \left(\frac{1}{3}\right)^0 \][/tex]
Knowing that any number to the power of 0 is 1:
[tex]\[ f(0) = 3 \times 1 = 3 \][/tex]
Therefore, the statement "The initial value of the function is [tex]\(\frac{1}{3}\)[/tex]" is false. The correct initial value is 3.
2. Base of the Function:
- The base of the exponential function [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] is the part raised to the power [tex]\( x \)[/tex]. Here, the base is [tex]\(\frac{1}{3}\)[/tex], which is visible directly within the function’s expression. Thus the statement "The base of the function is [tex]\(\frac{1}{3}\)[/tex]" is true.
3. Exponential Decay:
- Exponential functions can either show decay or growth. The function exhibits exponential decay if the base [tex]\( b \)[/tex] satisfies [tex]\( 0 < b < 1 \)[/tex]. Since the base [tex]\(\frac{1}{3}\)[/tex] is between 0 and 1:
[tex]\[ 0 < \frac{1}{3} < 1 \][/tex]
The function indeed shows exponential decay. Therefore, the statement "The function shows exponential decay" is true.
4. Stretch of the Function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex]:
- Stretching a function vertically involves multiplying it by a constant factor. Comparing [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] and [tex]\( g(x) = \left(\frac{1}{3}\right)^x \)[/tex], we notice that [tex]\( f(x) \)[/tex] is obtained by multiplying [tex]\( g(x) \)[/tex] by 3, indicating a vertical stretch by a factor of 3. Hence, the statement "The function is a stretch of the function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex]" is true.
5. Shrink of the Function [tex]\( f(x) = 3^x \)[/tex]:
- Considering the function [tex]\( f(x) = 3^x \)[/tex], if you take the reciprocal of the base ([tex]\(3\)[/tex]) and raise it to [tex]\( x \)[/tex], you get [tex]\(\left(\frac{1}{3}\right)^x\)[/tex]. Therefore, [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] can be considered a shrink (specifically horizontal) of the function [tex]\( f(x) = 3^x \)[/tex]. So, the statement "The function is a shrink of the function [tex]\( f(x) = 3^x \)[/tex]" is true.
Summarizing, the three correct statements about the function [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] and its graph 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].
- The function is a shrink of the function [tex]\( f(x) = 3^x \)[/tex].
1. Initial Value of the Function:
- The initial value of [tex]\( f(x) \)[/tex] is found by evaluating the function at [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3 \left(\frac{1}{3}\right)^0 \][/tex]
Knowing that any number to the power of 0 is 1:
[tex]\[ f(0) = 3 \times 1 = 3 \][/tex]
Therefore, the statement "The initial value of the function is [tex]\(\frac{1}{3}\)[/tex]" is false. The correct initial value is 3.
2. Base of the Function:
- The base of the exponential function [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] is the part raised to the power [tex]\( x \)[/tex]. Here, the base is [tex]\(\frac{1}{3}\)[/tex], which is visible directly within the function’s expression. Thus the statement "The base of the function is [tex]\(\frac{1}{3}\)[/tex]" is true.
3. Exponential Decay:
- Exponential functions can either show decay or growth. The function exhibits exponential decay if the base [tex]\( b \)[/tex] satisfies [tex]\( 0 < b < 1 \)[/tex]. Since the base [tex]\(\frac{1}{3}\)[/tex] is between 0 and 1:
[tex]\[ 0 < \frac{1}{3} < 1 \][/tex]
The function indeed shows exponential decay. Therefore, the statement "The function shows exponential decay" is true.
4. Stretch of the Function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex]:
- Stretching a function vertically involves multiplying it by a constant factor. Comparing [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] and [tex]\( g(x) = \left(\frac{1}{3}\right)^x \)[/tex], we notice that [tex]\( f(x) \)[/tex] is obtained by multiplying [tex]\( g(x) \)[/tex] by 3, indicating a vertical stretch by a factor of 3. Hence, the statement "The function is a stretch of the function [tex]\( f(x) = \left(\frac{1}{3}\right)^x \)[/tex]" is true.
5. Shrink of the Function [tex]\( f(x) = 3^x \)[/tex]:
- Considering the function [tex]\( f(x) = 3^x \)[/tex], if you take the reciprocal of the base ([tex]\(3\)[/tex]) and raise it to [tex]\( x \)[/tex], you get [tex]\(\left(\frac{1}{3}\right)^x\)[/tex]. Therefore, [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] can be considered a shrink (specifically horizontal) of the function [tex]\( f(x) = 3^x \)[/tex]. So, the statement "The function is a shrink of the function [tex]\( f(x) = 3^x \)[/tex]" is true.
Summarizing, the three correct statements about the function [tex]\( f(x) = 3 \left(\frac{1}{3}\right)^x \)[/tex] and its graph 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].
- The function is a shrink of the function [tex]\( f(x) = 3^x \)[/tex].
