To understand the relationship between the two functions [tex]\( f(x) = 0.7(6)^x \)[/tex] and [tex]\( g(x) = 0.7(6)^{-x} \)[/tex], let's analyze the given expressions step-by-step.
### Step 1: Understanding the Exponents
- Function [tex]\( f(x) \)[/tex]: [tex]\( f(x) = 0.7(6)^x \)[/tex]
- Here, [tex]\( f(x) \)[/tex] is an exponential function where the base is 6 and the exponent is [tex]\( x \)[/tex]. This means as [tex]\( x \)[/tex] increases, [tex]\( (6)^x \)[/tex] increases exponentially, and thus [tex]\( f(x) \)[/tex] grows very quickly.
- Function [tex]\( g(x) \)[/tex]: [tex]\( g(x) = 0.7(6)^{-x} \)[/tex]
- In this case, the exponent is [tex]\( -x \)[/tex]. This transformation can be understood by recalling that [tex]\( (6)^{-x} = \frac{1}{(6)^x} \)[/tex]. Therefore, [tex]\( g(x) = 0.7 \cdot \frac{1}{(6)^x} \)[/tex], which simplifies to [tex]\( g(x) = 0.7 \cdot (6^{-x}) \)[/tex].
### Step 2: Graphical Interpretation
- When you replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in the exponent, it reflects the graph of the original function over the y-axis.
### Step 3: Visualizing Reflections
- Reflection over the y-axis: If [tex]\( g(x) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] over the y-axis, for every point [tex]\((x, f(x))\)[/tex] on the graph of [tex]\( f(x) \)[/tex], there will be a corresponding point [tex]\((-x, f(x))\)[/tex] on the graph of [tex]\( g(x) \)[/tex]. This means [tex]\( g(x) \)[/tex] will be identical to [tex]\( f(x) \)[/tex], but mirrored along the y-axis.
### Conclusion
The function [tex]\( g(x) = 0.7(6)^{-x} \)[/tex] is indeed the reflection of [tex]\( f(x) \)[/tex] over the y-axis. Therefore, the relationship between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] can be described as follows:
[tex]\[ g(x) = 0.7(6)^{-x} \text{ is the reflection of } f(x) = 0.7(6)^x \text{ over the y-axis.} \][/tex]
Thus, the correct answer is:
[tex]\[ g(x) \text{ is the reflection of } f(x) \text{ over the } y\text{-axis}. \][/tex]
