a. The function $f$ is exponential and the points $(0,60)$ and $(1,81)$ are on the graph of $f$. (Draw a diagram of this!) Write a function formula for $f$.
\[
f(x)=60(1.35)^{\wedge} x \quad \text { Preview } 60(1.35)^{x} \text { syntax ok }
\]
b. The function $g$ is exponential and the points $(1,3.22)$ and $(2,7.406)$ are on the graph of $f$. (Draw a diagram of this!) Write a function formula for $g$.
\[
g(x)=
\]
Preview
Enter an algebraic expression [more..]
Answer
Finally, we check our results. For \(f\), when \(x = 0\), \(f(x) = 60(1.35)^0 = 60\), and when \(x = 1\), \(f(x) = 60(1.35)^1 = 81\). For \(g\), when \(x = 1\), \(g(x) = 3.22(2.3)^0 = 3.22\), and when \(x = 2\), \(g(x) = 3.22(2.3)^1 = 7.406\). Both functions meet the requirements of the problem.
Step 1 ：First, we need to understand that an exponential function has the form \(f(x) = ab^x\), where \(a\) is the initial value (the value of \(f\) when \(x = 0\)), and \(b\) is the growth factor.
Step 2 ：For the function \(f\), we know that when \(x = 0\), \(f(x) = 60\). So, \(a = 60\).
Step 3 ：We also know that when \(x = 1\), \(f(x) = 81\). We can use this information to find \(b\). We have the equation \(60b = 81\). Solving for \(b\), we get \(b = 81/60 = 1.35\).
Step 4 ：So, the function \(f\) is \(f(x) = 60(1.35)^x\).
Step 5 ：For the function \(g\), we follow the same steps. We know that when \(x = 1\), \(g(x) = 3.22\). So, \(a = 3.22\).
Step 6 ：We also know that when \(x = 2\), \(g(x) = 7.406\). We can use this information to find \(b\). We have the equation \(3.22b = 7.406\). Solving for \(b\), we get \(b = 7.406/3.22 = 2.3\).
Step 7 ：So, the function \(g\) is \(g(x) = 3.22(2.3)^{x-1}\).
Step 8 ：Finally, we check our results. For \(f\), when \(x = 0\), \(f(x) = 60(1.35)^0 = 60\), and when \(x = 1\), \(f(x) = 60(1.35)^1 = 81\). For \(g\), when \(x = 1\), \(g(x) = 3.22(2.3)^0 = 3.22\), and when \(x = 2\), \(g(x) = 3.22(2.3)^1 = 7.406\). Both functions meet the requirements of the problem.
