Consider the exponential function [tex]$f(x)=3\left(\frac{1}{3}\right)^x$[/tex] and its graph.
Which statements are true for this function and graph? Select three options.
* The initial value of the function is [tex]$\frac{1}{3}$[/tex].
* 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].
Answer (2)
The initial value is found by evaluating f ( 0 ) , which equals 3.
The base of the exponential function f ( x ) = 3 \tleft ( 3 1 \tright ) x is 3 1 .
Since the base 3 1 is between 0 and 1, the function shows exponential decay.
The function is a vertical stretch of f ( x ) = ( 3 1 ) x by a factor of 3.
The final answer is: The base of the function is 3 1 , the function shows exponential decay, and the function is a stretch of the function f ( x ) = ( 3 1 ) x .
Explanation
Analyzing the Problem Let's analyze the given exponential function and determine which statements are true. The function is given by f ( x ) = 3 \tleft ( 3 1 \tright ) x . We need to check each statement individually.
Checking the Initial Value The initial value of the function is the value of f ( x ) when x = 0 . So, we need to calculate f ( 0 ) .
f ( 0 ) = 3 \tleft ( 3 1 \tright ) 0 = 3 \tcdot 1 = 3 Thus, the initial value of the function is 3, not 3 1 . So, the first statement is false.
Identifying the Base The base of the exponential function is the number that is raised to the power of x . In the given function f ( x ) = 3 \tleft ( 3 1 \tright ) x , the base is 3 1 . So, the second statement is true.
Determining Exponential Decay or Growth To determine if the function shows exponential decay or growth, we look at the base. If the base is between 0 and 1, the function shows exponential decay. If the base is greater than 1, it shows exponential growth. Since the base is 3 1 , which is between 0 and 1, the function shows exponential decay. So, the third statement is true.
Checking for Vertical Stretch Now, let's check if the function is a stretch of the function f ( x ) = \tleft ( 3 1 \tright ) x . A vertical stretch occurs when we multiply the function by a constant greater than 1. In our case, f ( x ) = 3 \tleft ( 3 1 \tright ) x , which is the function ( 3 1 ) x multiplied by 3. Since 3 is greater than 1, the function is a vertical stretch of f ( x ) = ( 3 1 ) x . So, the fourth statement is true.
Checking for Vertical Shrink Finally, let's check if the function is a shrink of the function f ( x ) = 3 x . We can rewrite the given function as follows: f ( x ) = 3 \tleft ( 3 1 \tright ) x = 3 ( 3 − 1 ) x = 3 ( 3 − x ) = 3 1 − x This is not a simple shrink of f ( x ) = 3 x . A shrink would be of the form c \tcdot 3 x where 0 < c < 1 . So, the fifth statement is false.
Final Answer Therefore, the true statements are:
The base of the function is 3 1 .
The function shows exponential decay.
The function is a stretch of the function f ( x ) = ( 3 1 ) x .
Examples
Exponential functions are incredibly useful in modeling real-world phenomena. For instance, they can describe the decay of radioactive substances, the growth of populations, or the change in temperature of an object over time. Imagine you're tracking the depreciation of a car's value. If the car loses 1/3 of its value each year, and it initially costs 15 , 000 , t h e f u n c t i o n V(t) = 15000 \left(\frac{2}{3}\right)^t m o d e l s i t s v a l u e V(t) a f t er t$ years. Understanding exponential functions helps you predict future values and make informed decisions.
The correct statements about the function f ( x ) = 3 ( 3 1 ) x are that the base is 3 1 , it shows exponential decay, and it is a stretch of the function f ( x ) = ( 3 1 ) x .
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