How are the two functions [tex]$f(x)=0.7(6)^x$[/tex] and [tex]$g(x)=0.7(6)^{-x}$[/tex] related to each other?
A. [tex][tex]$g(x)$[/tex][/tex] is the reflection of [tex]$f(x)$[/tex] over the x-axis. B. [tex]$g(x)$[/tex] is the reflection of [tex][tex]$f(x)$[/tex][/tex] over the y-axis. C. [tex]$g(x)$[/tex] is the reflection of [tex]$f(x)$[/tex] over both axes. D. [tex][tex]$g(x)$[/tex][/tex] and [tex]$f(x)$[/tex] will appear to be the same function.
To determine the relationship between the two functions [tex]\( f(x) = 0.7 \cdot 6^x \)[/tex] and [tex]\( g(x) = 0.7 \cdot 6^{-x} \)[/tex], let's analyze them step by step.
### Step 1: Understand [tex]\( f(x) \)[/tex]
The function [tex]\( f(x) = 0.7 \cdot 6^x \)[/tex] is an exponential function where the base 6 is raised to the power of [tex]\( x \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( 6^x \)[/tex] grows rapidly because 6 is greater than 1, hence [tex]\( f(x) \)[/tex] will also grow rapidly since it is a positive constant (0.7) times an exponentially growing term.
### Step 2: Understand [tex]\( g(x) \)[/tex]
The function [tex]\( g(x) = 0.7 \cdot 6^{-x} \)[/tex] is also an exponential function; however, the exponent here is the negative of [tex]\( x \)[/tex]. This can be rewritten as: [tex]\[ g(x) = 0.7 \cdot \frac{1}{6^x} \][/tex] Since [tex]\( 6^x \)[/tex] grows rapidly as [tex]\( x \)[/tex] increases, [tex]\(\frac{1}{6^x}\)[/tex] will decrease rapidly as [tex]\( x \)[/tex] increases. Therefore, [tex]\( g(x) \)[/tex] will decrease rapidly as [tex]\( x \)[/tex] increases.
### Step 3: Recognize the Reflection Over the y-Axis
The function [tex]\( g(x) = 0.7 \cdot 6^{-x} \)[/tex] can be interpreted as a reflection of [tex]\( f(x) \)[/tex] over the [tex]\( y \)[/tex]-axis because replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in [tex]\( f(x) \)[/tex] gives [tex]\( g(x) \)[/tex].
Here is the reflection process step-by-step: - If we have [tex]\( f(x) = 0.7 \cdot 6^x \)[/tex] and reflect it over the [tex]\( y \)[/tex]-axis, we replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex]: [tex]\[ f(-x) = 0.7 \cdot 6^{-x} \][/tex] - This is exactly the function [tex]\( g(x) \)[/tex].
### Conclusion
The function [tex]\( g(x) = 0.7 \cdot 6^{-x} \)[/tex] is the reflection of [tex]\( f(x) = 0.7 \cdot 6^x \)[/tex] over the [tex]\( y \)[/tex]-axis because we have [tex]\( g(x) = f(-x) \)[/tex].
Therefore, the correct relationship is: [tex]\[ g(x) \text{ is the reflection of } f(x) \text{ over the } y\text{-axis.} \][/tex]
### Answer
[tex]\( g(x) \)[/tex] is the reflection of [tex]\( f(x) \)[/tex] over the [tex]\( y \)[/tex]-axis.
