Find solutions to your questions with the help of IDNLearn.com's expert community. Discover comprehensive answers from knowledgeable members of our community, covering a wide range of topics to meet all your informational needs.
Sagot :
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}
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}
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is committed to your satisfaction. Thank you for visiting, and see you next time for more helpful answers.