The function [tex]\( f(x) = 3 \left( \frac{2}{3} \right)^x \)[/tex] is an exponential function. Let’s analyze its components and characteristics.
1. General Form of an Exponential Function: The general form of an exponential function can be written as [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is the initial value (or the y-intercept when [tex]\( x = 0 \)[/tex]) and [tex]\( b \)[/tex] is the base of the exponential function.
2. Initial Value: In our function, [tex]\( a = 3 \)[/tex]. So, when [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3 \left( \frac{2}{3} \right)^0 = 3 \cdot 1 = 3
\][/tex]
Thus, the y-intercept of the function is [tex]\( (0, 3) \)[/tex].
3. Base Value: Here, the base [tex]\( b = \frac{2}{3} \)[/tex]:
- Since [tex]\( \frac{2}{3} < 1 \)[/tex], this indicates that we have an exponential decay function. This means the function will decrease as [tex]\( x \)[/tex] increases.
4. Behavior as [tex]\( x \)[/tex] Approaches Infinity and Negative Infinity:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( \left( \frac{2}{3} \right)^x \rightarrow 0 \)[/tex]. Therefore, [tex]\( f(x) \rightarrow 3 \cdot 0 = 0 \)[/tex]. The graph approaches the x-axis but never touches it.
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( \left( \frac{2}{3} \right)^x \rightarrow \infty \)[/tex]. Therefore, [tex]\( f(x) \rightarrow 3 \cdot \infty = \infty \)[/tex]. This means as [tex]\( x \)[/tex] gets very negative, the function’s value becomes very large.
5. Plotting Key Points:
- Let’s find a few more points to plot the graph.
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3 \left( \frac{2}{3} \right)^1 = 3 \cdot \frac{2}{3} = 2
\][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 3 \left( \frac{2}{3} \right)^{-1} = 3 \cdot \frac{3}{2} = 4.5
\][/tex]
6. Graph Characteristics:
- The graph will pass through the points [tex]\((0, 3)\)[/tex], [tex]\((1, 2)\)[/tex], and [tex]\((-1, 4.5)\)[/tex].
- The function decreases as [tex]\( x \)[/tex] increases, confirming it's an exponential decay function.
- The x-axis ([tex]\( y = 0 \)[/tex]) acts as a horizontal asymptote.
Combining these analysis, we can conclude:
The graph of [tex]\( f(x)=3\left( \frac{2}{3} \right)^x \)[/tex] is an exponential decay function.
