To determine the exponential function [tex]\( f(x) \)[/tex] given in the table, we need to find the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that [tex]\( f(x) = a \cdot b^x \)[/tex].
Let's look at the given points from the table:
[tex]\[
\begin{array}{cc}
x & f(x) \\
\hline
0 & 9 \\
1 & 15 \\
\end{array}
\][/tex]
We can use these points to solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
1. Determine [tex]\( a \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = a \cdot b^0 \)[/tex]. We know that [tex]\( b^0 = 1 \)[/tex], so:
[tex]\[
f(0) = a \cdot 1 \Rightarrow a = f(0) = 9
\][/tex]
Hence, [tex]\( a = 9 \)[/tex].
2. Determine [tex]\( b \)[/tex]:
- Use the point [tex]\( (1, 15) \)[/tex]. When [tex]\( x = 1 \)[/tex], we have:
[tex]\[
f(1) = 9 \cdot b^1 \Rightarrow 15 = 9 \cdot b
\][/tex]
Solving for [tex]\( b \)[/tex], we divide both sides by 9:
[tex]\[
b = \frac{15}{9} = \frac{5}{3} \approx 1.6666666666666667
\][/tex]
Using the calculated values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can now write the complete exponential function:
[tex]\[
f(x) = 9 \cdot \left(\frac{5}{3}\right)^x
\][/tex]
So the completed equation for [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = 9 \cdot 1.6666666666666667^x
\][/tex]
