To understand and analyze the function [tex]\( f(x) = 2 \cdot 3^x - 1 \)[/tex], let's take a look at it step-by-step:
### Step 1: Understanding the Function
The function [tex]\( f(x) = 2 \cdot 3^x - 1 \)[/tex] is an exponential function modified by a linear transformation. Here, [tex]\( 3^x \)[/tex] is an exponential term and it's being scaled by 2 and then shifted down by 1.
### Step 2: Evaluating the Function at Specific Points
Let's calculate the values of [tex]\( f(x) \)[/tex] at a few specific points to understand its behavior better.
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2 \cdot 3^0 - 1 = 2 \cdot 1 - 1 = 2 - 1 = 1
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2 \cdot 3^1 - 1 = 2 \cdot 3 - 1 = 6 - 1 = 5
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 2 \cdot 3^{-1} - 1 = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3}
\][/tex]
### Step 3: General Behavior Analysis
- Growth: Because the base [tex]\( 3 \)[/tex] of the exponential term [tex]\( 3^x \)[/tex] is greater than 1, the term [tex]\( 3^x \)[/tex] grows rapidly as [tex]\( x \)[/tex] increases. Consequently, [tex]\( f(x) = 2 \cdot 3^x - 1 \)[/tex] also grows rapidly as [tex]\( x \)[/tex] increases.
- Decay: For negative values of [tex]\( x \)[/tex], [tex]\( 3^x \)[/tex] becomes a fraction, approaching 0 as [tex]\( x \)[/tex] decreases further. This makes [tex]\( f(x) \)[/tex] approach the value of [tex]\(-1\)[/tex], but never actually reaching [tex]\(-1\)[/tex].
### Step 4: Asymptotic Behavior
As [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( 3^x \)[/tex] approaches 0, and thus:
[tex]\[
f(x) = 2 \cdot 3^x - 1 \rightarrow 2 \cdot 0 - 1 = -1
\][/tex]
This tells us that [tex]\( y = -1 \)[/tex] is a horizontal asymptote of the function.
### Step 5: Intercepts
- Y-Intercept: The function intercepts the y-axis where [tex]\( x = 0 \)[/tex], which we already calculated to be:
[tex]\[
f(0) = 1
\][/tex]
- X-Intercept: To find the x-intercept, set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
2 \cdot 3^x - 1 = 0
\][/tex]
[tex]\[
2 \cdot 3^x = 1
\][/tex]
[tex]\[
3^x = \frac{1}{2}
\][/tex]
Taking the natural logarithm (ln) of both sides:
[tex]\[
x \cdot \ln(3) = \ln\left(\frac{1}{2}\right)
\][/tex]
[tex]\[
x = \frac{\ln\left(\frac{1}{2}\right)}{\ln(3)} = \frac{-\ln(2)}{\ln(3)}
\][/tex]
### Conclusion
The function [tex]\( f(x) = 2 \cdot 3^x - 1 \)[/tex] represents an exponential growth shifted down by 1. It has a y-intercept at [tex]\( f(0) = 1 \)[/tex], an x-intercept at [tex]\( x = \frac{-\ln(2)}{\ln(3)} \)[/tex], and a horizontal asymptote at [tex]\( y = -1 \)[/tex].
