To understand why the initial value of any function of the form \( f(x) = a \left(b^x\right) \) is equal to \( a \), let's go through the process step-by-step:
### Step 1: Define the Function
We start with the given function:
[tex]\[ f(x) = a \left(b^x\right) \][/tex]
where:
- \( a \) is a coefficient,
- \( b \) is the base of the exponential term,
- \( x \) is the variable.
### Step 2: Evaluate the Function at \( x = 0 \)
The initial value of the function refers to the value of the function when \( x = 0 \).
So, we substitute \( x = 0 \) into the function:
[tex]\[ f(0) = a \left(b^0\right) \][/tex]
### Step 3: Simplify the Exponential Term
Next, we need to simplify the term \( b^0 \). By definition of exponents, any nonzero number raised to the power of 0 is equal to 1:
[tex]\[ b^0 = 1 \][/tex]
### Step 4: Calculate the Initial Value
Now, substitute \( b^0 \) with 1 in the equation:
[tex]\[ f(0) = a \left(1\right) \][/tex]
Since any number multiplied by 1 is that number itself, we get:
[tex]\[ f(0) = a \][/tex]
### Conclusion
Therefore, the initial value of the function \( f(x) = a \left(b^x\right) \) when \( x = 0 \) is indeed \( a \).
This concludes our derivation showing that the initial value ([tex]\( f(0) \)[/tex]) of the function [tex]\( f(x) = a \left(b^x\right) \)[/tex] is equal to [tex]\( a \)[/tex].
