To determine which functions are exponential and identify their bases, let's analyze each given function.
1. [tex]\( f(x) = 4e^x \)[/tex]
- Is the function an exponential function? Yes, this is an exponential function because it can be written in the form [tex]\(a \cdot b^x\)[/tex], where [tex]\(a = 4\)[/tex] and [tex]\(b = e\)[/tex].
- Identify the base. The base is [tex]\(e\)[/tex].
2. [tex]\( h(x) = -2x^2 \)[/tex]
- Is the function an exponential function? No, this is not an exponential function. It's actually a polynomial function of degree 2 because it's based on [tex]\(x^2\)[/tex].
- Identify the base. There is no base because it is not an exponential function.
3. [tex]\( f(t) = -2(1.05)^{4t} \)[/tex]
- Is the function an exponential function? Yes, this is an exponential function. It can be written in the form [tex]\(a \cdot b^{ct}\)[/tex], where [tex]\(a = -2\)[/tex], [tex]\(b = 1.05\)[/tex], and [tex]\(c = 4\)[/tex]. The variable [tex]\(t\)[/tex] is in the exponent.
- Identify the base. The base is [tex]\(1.05\)[/tex].
4. [tex]\( v(r) = \frac{4}{3}\pi r^3 \)[/tex]
- Is the function an exponential function? No, this is not an exponential function. It's a polynomial function of degree 3 because it involves [tex]\(r^3\)[/tex].
- Identify the base. There is no base because it is not an exponential function.
Now, let's put the answers into the table format:
\begin{tabular}{|l|l|l|}
\hline
Functions & \begin{tabular}{l}
Is the function an exponential \\
function?
\end{tabular} & Identify the base. \\
\hline
[tex]$f(x)=4 e^x$[/tex] & Yes & [tex]$e$[/tex] \\
\hline
[tex]$h(x)=-2 x^2$[/tex] & No & None \\
\hline
[tex]$f(t)=-2(1.05)^{4 t}$[/tex] & Yes & 1.05 \\
\hline
[tex]$v(r)=\frac{4}{3} \pi r^3$[/tex] & No & None \\
\hline
\end{tabular}
