To determine the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the exponential function [tex]\(f(x) = a \cdot b^x\)[/tex], we can use the given values from the table. Here's the step-by-step procedure:
### Step 1: Using the value at [tex]\(x = 0\)[/tex]
From the table, when [tex]\(x = 0\)[/tex], [tex]\(f(x) = 5\)[/tex]. We can write this as:
[tex]\[ f(0) = a \cdot b^0 = a \cdot 1 = a \][/tex]
So,
[tex]\[ a = 5 \][/tex]
### Step 2: Using the value at [tex]\(x = 1\)[/tex]
From the table, when [tex]\(x = 1\)[/tex], [tex]\(f(x) = 8\)[/tex]. Substituting [tex]\(a = 5\)[/tex] and [tex]\(x = 1\)[/tex] into the function gives:
[tex]\[ f(1) = a \cdot b^1 = 5 \cdot b \][/tex]
According to the table, [tex]\(f(1) = 8\)[/tex]. Therefore:
[tex]\[ 8 = 5 \cdot b \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = \frac{8}{5} \][/tex]
[tex]\[ b = 1.6 \][/tex]
### Putting it all together
Now that we have the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we can write the completed equation for the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 5 \cdot 1.6^x \][/tex]
So, the completed equation for [tex]\(f(x)\)[/tex] is:
[tex]\[ f(x) = 5 \cdot 1.6^x \][/tex]
