To write the exponential function that passes through the points (1,6) and (2,12), we will follow a detailed step-by-step solution:
### Step 1: Solve for [tex]\( b \)[/tex] by dividing consecutive function values
The function values at the given points are:
[tex]\[ f(1) = 6 \][/tex]
[tex]\[ f(2) = 12 \][/tex]
We need to solve for [tex]\( b \)[/tex] using the relationship:
[tex]\[ b = \frac{f(x+1)}{f(x)} \][/tex]
Substituting the given values:
[tex]\[ b = \frac{f(2)}{f(1)} = \frac{12}{6} = 2 \][/tex]
So, [tex]\( b = 2 \)[/tex].
### Step 2: Use the exponential function equation [tex]\( f(x) = a b^x \)[/tex] to solve for [tex]\( a \)[/tex]
Using the point [tex]\((1, 6)\)[/tex]:
[tex]\[ f(x) = a b^x \][/tex]
[tex]\[ f(1) = a b^1 \][/tex]
[tex]\[ 6 = a \cdot 2^1 \][/tex]
[tex]\[ 6 = 2a \][/tex]
[tex]\[ a = \frac{6}{2} = 3 \][/tex]
Thus, [tex]\( a = 3 \)[/tex].
### Step 3: Write the exponential function [tex]\( f(x) = a b^x \)[/tex]
Substitute [tex]\( a = 3 \)[/tex] and [tex]\( b = 2 \)[/tex] into the exponential function equation:
So, the exponential function is:
[tex]\[ f(x) = 3 \cdot 2^x \][/tex]
