Certainly! Let's work with the function [tex]\( f(x) = 5 \cdot 3^x \)[/tex]. Let me provide you with a detailed, step-by-step explanation of how this function behaves.
### Step 1: Understanding the Function
The function [tex]\( f(x) = 5 \cdot 3^x \)[/tex] is an exponential function. In general terms, it multiplies the constant 5 by 3 raised to the power of [tex]\( x \)[/tex].
### Step 2: Values of the Function
To understand better how this function operates, let's calculate some values for specific inputs of [tex]\( x \)[/tex].
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 5 \cdot 3^0 = 5 \cdot 1 = 5
\][/tex]
2. When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 5 \cdot 3^1 = 5 \cdot 3 = 15
\][/tex]
3. When [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 5 \cdot 3^2 = 5 \cdot 9 = 45
\][/tex]
4. When [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} \approx 1.67
\][/tex]
### Step 3: Observations
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] grows exponentially.
- As [tex]\( x \)[/tex] decreases (negative values), [tex]\( f(x) \)[/tex] approaches zero but does not reach zero (assuming [tex]\( x \rightarrow -\infty \)[/tex]).
- The base of the exponent is 3, and its effect is scaled by the multiplier 5.
### Step 4: Graphical Representation
The graph of [tex]\( f(x) = 5 \cdot 3^x \)[/tex] will have the following characteristics:
- It always passes through the point [tex]\( (0, 5) \)[/tex].
- It increases rapidly for positive values of [tex]\( x \)[/tex].
- It approaches the x-axis but never touches it as [tex]\( x \rightarrow -\infty \)[/tex].
### Step 5: Asymptotic Behavior
- As [tex]\( x \rightarrow \infty \)[/tex]:
The function [tex]\( f(x) \)[/tex] increases towards infinity.
[tex]\[
\lim_{x \to \infty} 5 \cdot 3^x = \infty
\][/tex]
- As [tex]\( x \rightarrow -\infty \)[/tex]:
The function [tex]\( f(x) \)[/tex] asymptotically approaches zero.
[tex]\[
\lim_{x \to -\infty} 5 \cdot 3^x = 0
\][/tex]
### Conclusion
The function [tex]\( f(x) = 5 \cdot 3^x \)[/tex] is an exponential function characterized by growing rapidly for positive [tex]\( x \)[/tex] and decaying towards zero for negative [tex]\( x \)[/tex]. The constant 5 serves as the scaling factor, and the base 3 dictates the rate of exponential growth.
