At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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].
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.