What are the domain and range of the function?
| x | y |
|---|--------|
| 0 | 4 |
| 1 | 5 |
| 2 | 6.25 |
| 3 | 7.8125 |
A. The domain is the set of integers, and the range is [tex]$y \ge 4$[/tex].
B. The domain is the set of real numbers, and the range is [tex]$y \ge 0$[/tex].
C. The domain is the set of integers, and the range is [tex]$y > 4$[/tex].
D. The domain is the set of real numbers, and the range is [tex]$y > 4$[/tex].
Answer (2)
Determine the exponential function: y = a ⋅ b x , where a = 4 and b = 1.25 , so y = 4 ⋅ ( 1.25 ) x .
The domain of a continuous exponential function is all real numbers.
The range of y = 4 ⋅ ( 1.25 ) x is all 0"> y > 0 .
Therefore, the domain is the set of real numbers, and the range is 0}"> y > 0 .
Explanation
Understanding the Problem We are given a table of x and y values for a continuous exponential function and asked to determine its domain and range. The table provides the following ordered pairs: ( 0 , 4 ) , ( 1 , 5 ) , ( 2 , 6.25 ) , and ( 3 , 7.8125 ) .
Setting up Equations To find the exponential function that fits the given data points, we assume the function is of the form y = a ⋅ b x + c . Using the points ( 0 , 4 ) and ( 1 , 5 ) , we can set up a system of equations:
4 = a ⋅ b 0 + c and 5 = a ⋅ b 1 + c .
This simplifies to 4 = a + c and 5 = ab + c . From the first equation, we have c = 4 − a . Substituting this into the second equation, we get 5 = ab + 4 − a , which simplifies to 1 = ab − a = a ( b − 1 ) .
Solving for b Now, let's use the point ( 2 , 6.25 ) : 6.25 = a ⋅ b 2 + c = a ⋅ b 2 + 4 − a , so 2.25 = a ( b 2 − 1 ) = a ( b − 1 ) ( b + 1 ) . Since a ( b − 1 ) = 1 , we have 2.25 = ( b + 1 ) , which gives us b = 1.25 .
Solving for a and c Substituting b = 1.25 back into a ( b − 1 ) = 1 , we get a ( 1.25 − 1 ) = 1 , so 0.25 a = 1 , which means a = 4 . Finally, c = 4 − a = 4 − 4 = 0 . Thus, the exponential function is y = 4 ⋅ ( 1.25 ) x .
Determining Domain and Range Since the function is a continuous exponential function, the domain is all real numbers. The exponential term ( 1.25 ) x is always positive, so 4 ⋅ ( 1.25 ) x is always positive. Therefore, the range is 0"> y > 0 .
Final Answer The domain of the function is the set of all real numbers, and the range is 0"> y > 0 .
Examples
Exponential functions are incredibly useful in modeling real-world phenomena, such as population growth, radioactive decay, and compound interest. For instance, if a population of bacteria starts at 4 million and grows by 25% every unit of time, the function y = 4 ⋅ ( 1.25 ) x models the population size y after x time units. Understanding the domain and range helps us interpret the model's validity: time can be any real number (domain), and the population size will always be a positive value (range). This allows us to make predictions and analyze trends effectively.
The domain of the function is the set of integers {0, 1, 2, 3}, and the range of the function is all values y such that y ≥ 4. Thus, the correct answer is Option A. This means that the function accepts the integers 0 through 3 as inputs and outputs values that start from 4 and go upwards.
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