Q. 115 - Precalculus Enhanced with Graphing Utilities

Question

If f is an exponential function of the form $f\left(x\right)=C·a^x$ with growth factor $3$ and $f\left(6\right)=12$, what is $f\left(7\right)$?

Step-by-Step Solution

Answer

The value of $f\left(7\right)$ is $36$.

Step 1. Given information.

Consider the given question,

$$\begin{gathered} f\left(x\right)=C·a^x, \\ f\left(6\right)=12, \\ a=3 \end{gathered}$$

Substitute the values in the function,

$$\begin{gathered} f\left(x\right)=C·a^x \\ f\left(6\right)=C·a^6 \\ 12=C·{\left(3\right)}^6 \\ C=\frac{{\left(3\right)}^6}{12} \end{gathered}$$

Step 2. Substitute x = 7 in the function.

Consider the given question,

$$\begin{gathered} f\left(x\right)=C·a^x, \\ x=7, \\ a=3, \\ C=\frac{12}{{\left(3\right)}^6} \end{gathered}$$

Substitute the values in the function,

$$\begin{gathered} C\left(7\right)=C·a^7 \\ =\frac{12}{{\left(3\right)}^6}\times {\left(3\right)}^7 \\ =12\times 3 \\ =36 \end{gathered}$$