To plot the given functions, we need to:
- Define the functions \( f(x) = 3^x \), \( f(x) = -3^x \), \( f(x) = \left(\frac{1}{5}\right)^x \), and \( f(x) = -\left(\frac{1}{5}\right)^x \).
- Generate a range of x-values.
- Compute the corresponding y-values for each function.
- Plot the functions using a graphing library like Matplotlib.
We are given four functions to analyze:
a. \( f(x) = 3^x \)
b. \( f(x) = -3^x \)
c. \( f(x) = \left(\frac{1}{5}\right)^x \)
d. \( f(x) = -\left(\frac{1}{5}\right)^x \)
This is an exponential function with base 3. The graph of \( 3^x \) is an increasing function that passes through the point (0,1) and approaches 0 as \( x \) approaches negative infinity.
This is the negative of the exponential function \( 3^x \). The graph of \( -3^x \) is a decreasing function that passes through the point (0,-1) and approaches 0 as \( x \) approaches negative infinity, but it is reflected over the x-axis compared to \( 3^x \).
This is an exponential function with base \( \frac{1}{5} \). The graph of \( \left(\frac{1}{5}\right)^x \) is a decreasing function that passes through the point (0,1) and approaches 0 as \( x \) approaches positive infinity.
This is the negative of the exponential function \( \left(\frac{1}{5}\right)^x \). The graph of \( -\left(\frac{1}{5}\right)^x \) is an increasing function that passes through the point (0,-1) and approaches 0 as \( x \) approaches positive infinity, but it is reflected over the x-axis compared to \( \left(\frac{1}{5}\right)^x \).
For \( f(x) = 3^x \):
\[
\boxed{f(x) = 3^x}
\]
For \( f(x) = -3^x \):
\[
\boxed{f(x) = -3^x}
\]
For \( f(x) = \left(\frac{1}{5}\right)^x \):
\[
\boxed{f(x) = \left(\frac{1}{5}\right)^x}
\]
Answered
