Given the function, [tex]f(x)=4^{x+2}[/tex], what is the initial value of the function?
[tex]f(0)=\square[/tex]
Answer (2)
The initial value of the function f ( x ) = 4 x + 2 when evaluated at x = 0 is 16 . This value represents the output of the function at the starting point. Understanding the initial value is crucial for analyzing various mathematical models involving growth or changes over time.
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The problem asks for the initial value of the function f ( x ) = 4 x + 2 .
To find the initial value, we evaluate the function at x = 0 .
Substitute x = 0 into the function: f ( 0 ) = 4 0 + 2 = 4 2 .
Simplify the expression to find the initial value: 16 .
Explanation
Understanding the Problem We are given the function f ( x ) = 4 x + 2 and asked to find its initial value. The initial value of a function is the value of the function when x = 0 .
Substituting x=0 To find the initial value, we need to evaluate f ( 0 ) . We substitute x = 0 into the function:
Calculating f(0) f ( 0 ) = 4 0 + 2 = 4 2
Simplifying the Expression Now, we simplify 4 2 :
Final Answer 4 2 = 4 × 4 = 16 . Therefore, the initial value of the function is 16.
Examples
In the context of exponential growth, such as bacterial population, the function f ( x ) = 4 x + 2 could represent the number of bacteria at time x (in hours). The initial value, f ( 0 ) = 16 , would then represent the starting number of bacteria. Understanding initial values is crucial in modeling and predicting population sizes, which has applications in medicine, environmental science, and biotechnology. For example, if we start with 16 bacteria, after 1 hour we will have f ( 1 ) = 4 1 + 2 = 4 3 = 64 bacteria, and after 2 hours we will have f ( 2 ) = 4 2 + 2 = 4 4 = 256 bacteria. This shows how the initial value affects the entire growth trajectory.
