Which exponential function has an initial value of $2$?
$f(x)=2\left(3^x\right)$
$f(x)=3\left(2^x\right)$
| x | f ( x ) |
| --- | ------- |
| -2 | $\frac{1}{8}$ |
| -1 | $\frac{1}{4}$ |
Answer (2)
Calculate the initial value of f ( x ) = 2 ( 3 x ) by evaluating f ( 0 ) , which equals 2.
Calculate the initial value of f ( x ) = 3 ( 2 x ) by evaluating f ( 0 ) , which equals 3.
Determine the exponential function from the table to be f ( x ) = 2 1 ( 2 x ) and find its initial value f ( 0 ) = 2 1 .
The exponential function with an initial value of 2 is f ( x ) = 2 ( 3 x ) .
Explanation
Understanding the Problem We are given two exponential functions and a table of values representing another exponential function. We need to determine which of these has an initial value of 2. The initial value of a function f ( x ) is the value of the function when x = 0 , i.e., f ( 0 ) .
Initial Value of the First Function Let's find the initial value of the first function, f ( x ) = 2 ( 3 x ) . We evaluate f ( 0 ) : f ( 0 ) = 2 ( 3 0 ) = 2 ( 1 ) = 2
Initial Value of the Second Function Now let's find the initial value of the second function, f ( x ) = 3 ( 2 x ) . We evaluate f ( 0 ) : f ( 0 ) = 3 ( 2 0 ) = 3 ( 1 ) = 3
Finding the Exponential Function from the Table Next, we need to determine the exponential function that passes through the points ( − 2 , 8 1 ) and ( − 1 , 4 1 ) . We assume the function is of the form f ( x ) = a b x . We can substitute the given points into the equation to create a system of two equations with two unknowns, a and b :
8 1 = a b − 2 4 1 = a b − 1
Solving for b Dividing the second equation by the first equation, we get: 8 1 4 1 = a b − 2 a b − 1 2 = b
Solving for a Now we can substitute b = 2 into either equation to solve for a . Using the second equation: 4 1 = a ( 2 ) − 1 4 1 = 2 a a = 2 1
Initial Value of the Function from the Table So the exponential function represented by the table is f ( x ) = 2 1 ( 2 x ) . Now we find the initial value of this function by evaluating f ( 0 ) :
f ( 0 ) = 2 1 ( 2 0 ) = 2 1 ( 1 ) = 2 1
Comparing Initial Values Comparing the initial values, we found that the first function, f ( x ) = 2 ( 3 x ) , has an initial value of 2. The second function has an initial value of 3, and the function represented by the table has an initial value of 2 1 .
Final Answer Therefore, the exponential function with an initial value of 2 is f ( x ) = 2 ( 3 x ) .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population starts with 2 individuals and triples every year, the population after x years can be modeled by the exponential function f ( x ) = 2 ( 3 x ) . This function tells us how the population grows over time, starting from an initial value of 2.
The exponential function with an initial value of 2 is f ( x ) = 2 ( 3 x ) since evaluating this function at x = 0 yields 2. The second function, f ( x ) = 3 ( 2 x ) , has an initial value of 3. Thus, the correct choice is the first function.
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