To understand how the graph represents the function [tex]\( f(x) = \left(\frac{3}{2}\right)^{-x} \)[/tex], let's follow a step-by-step analysis of this function.
1. Rewrite the Function: The negative exponent can be rewritten using the property of exponents, [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]. Thus, we can rewrite [tex]\( f(x) \)[/tex] as:
[tex]\[
f(x) = \left(\frac{3}{2}\right)^{-x} = \left(\frac{1}{\frac{3}{2}}\right)^x = \left(\frac{2}{3}\right)^x
\][/tex]
2. Interpret the Function: Now, the function [tex]\( f(x) = \left(\frac{2}{3}\right)^x \)[/tex] describes an exponential decay function. Key characteristics of the function [tex]\( f(x) = a^x \)[/tex] where [tex]\( 0 < a < 1 \)[/tex] are:
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases.
- As [tex]\( x \)[/tex] decreases, [tex]\( f(x) \)[/tex] increases.
- The function has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
- For [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 1 \)[/tex].
3. Behavior at Specific Points:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = \left(\frac{2}{3}\right)^0 = 1
\][/tex]
- For [tex]\( x > 0 \)[/tex], [tex]\( \left(\frac{2}{3}\right)^x \)[/tex] will be a fraction (less than 1) that gets smaller as [tex]\( x \)[/tex] gets larger.
- For [tex]\( x < 0 \)[/tex], [tex]\( \left(\frac{2}{3}\right)^x \)[/tex] will be a number greater than 1 and will grow larger as [tex]\( x \)[/tex] becomes more negative.
4. Graph Features:
- Intersects the y-axis at (0, 1): Always passes through the point (0, 1).
- Horizontal Asymptote: As [tex]\( x \)[/tex] approaches infinity, the function approaches 0, but never reaches 0.
- Decreasing Function: The function decreases as [tex]\( x \)[/tex] increases, since [tex]\( \frac{2}{3} \)[/tex] is a fraction less than 1.
5. Graph Example: Consider the following sample points:
- [tex]\( f(0) = 1 \)[/tex]
- [tex]\( f(1) = \left(\frac{2}{3}\right)^1 = \frac{2}{3} \)[/tex]
- [tex]\( f(-1) = \left(\frac{2}{3}\right)^{-1} = \left(\frac{3}{2}\right)^1 = \frac{3}{2} = 1.5 \)[/tex]
- [tex]\( f(2) = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \)[/tex]
Using these points and the characteristic behavior described, we can accurately plot this function.
The Graph of [tex]\( f(x) = \left( \frac{2}{3} \right)^x \)[/tex] has the following properties:
- It starts at (0, 1) and decreases towards the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] increases.
- As [tex]\( x \)[/tex] decreases (moving left on the graph), the function increases beyond 1.
Hence, when choosing from different graphs presented, look for one that matches these described properties.
