Identify the characteristics of linear and exponential functions.
| Function | Type and Behavior |
| ----------------- | ----------------- |
| [tex]p(t)=160(1.1)^t[/tex] | Select an answer |
| [tex]g(x)=-19 x+180[/tex] | Select an answer |
| [tex]d(y)=9(3)^y[/tex] | Select an answer |
| [tex]f(x)=34 x-230[/tex] | Select an answer |
| [tex]h(x)=220(0.8)^x[/tex] | Select an answer |
Answer (2)
p ( t ) = 160 ( 1.1 ) t is exponential and increasing.
g ( x ) = − 19 x + 180 is linear and decreasing.
d ( y ) = 9 ( 3 ) y is exponential and increasing.
f ( x ) = 34 x − 230 is linear and increasing.
h ( x ) = 220 ( 0.8 ) x is exponential and decreasing.
Explanation
Analyzing the Problem We are given a table of functions and we need to identify each function as either linear or exponential and determine whether it is increasing or decreasing. Let's analyze each function individually.
Analyzing p(t)
p ( t ) = 160 ( 1.1 ) t : This is an exponential function because it has the form c × b t , where c = 160 and b = 1.1 . Since 1"> b > 1 , the function is increasing.
Analyzing g(x)
g ( x ) = − 19 x + 180 : This is a linear function because it has the form a x + b , where a = − 19 and b = 180 . Since a < 0 , the function is decreasing.
Analyzing d(y)
d ( y ) = 9 ( 3 ) y : This is an exponential function because it has the form c × b y , where c = 9 and b = 3 . Since 1"> b > 1 , the function is increasing.
Analyzing f(x)
f ( x ) = 34 x − 230 : This is a linear function because it has the form a x + b , where a = 34 and b = − 230 . Since 0"> a > 0 , the function is increasing.
Analyzing h(x)
h ( x ) = 220 ( 0.8 ) x : This is an exponential function because it has the form c × b x , where c = 220 and b = 0.8 . Since 0 < b < 1 , the function is decreasing.
Examples
Understanding the behavior of linear and exponential functions is crucial in many real-world applications. For example, when modeling population growth, exponential functions can help predict how a population will increase over time. In contrast, linear functions can be used to model the depreciation of an asset, where the value decreases by a fixed amount each year. Recognizing these patterns allows for informed decision-making in finance, biology, and other fields. For instance, if a population grows by 10% each year, the function is exponential and increasing. If a car loses $2000 in value each year, the function is linear and decreasing.
The functions can be categorized as follows: p ( t ) and d ( y ) are exponential and increasing, g ( x ) is linear and decreasing, f ( x ) is linear and increasing, and h ( x ) is exponential and decreasing.
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