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]g(x)[/tex] is the reflection of [tex]f(x)[/tex] over the [tex]x[/tex]-axis.
B. [tex]g(x)[/tex] is the reflection of [tex]f(x)[/tex] over the [tex]y[/tex]-axis.
C. [tex]g(x)[/tex] is the reflection of [tex]f(x)[/tex] over both axes.
D. [tex]g(x)[/tex] and [tex]f(x)[/tex] will appear to be the same function.



Answer :

To determine the relationship between the two functions \( f(x) = 0.7 \cdot 6^x \) and \( g(x) = 0.7 \cdot 6^{-x} \), let's analyze them step-by-step.

### Step 1: Understand the Base Functions

1. Function \( f(x) \):
- The function \( f(x) = 0.7 \cdot 6^x \) is an exponential function where the base is \( 6 \) and \( x \) is the exponent.
- Exponential functions of the form \( a^x \) grow rapidly as \( x \) increases if \( a > 1 \).

2. Function \( g(x) \):
- The function \( g(x) = 0.7 \cdot 6^{-x} \) can be rewritten for better understanding as \( g(x) = 0.7 \cdot \frac{1}{6^x} \).
- This is an exponential function where the base is the reciprocal of \( 6 \), which causes the function to decay as \( x \) increases.

### Step 2: Relationship Analysis

To determine their relationship, let's consider the transformation involved:

- For \( f(x) = 0.7 \cdot 6^x \), when we replace \( x \) with \(-x\):
[tex]\[ f(-x) = 0.7 \cdot 6^{-x} \][/tex]
We see that \( g(x) = f(-x) \).

### Step 3: Identifying the Transformation

Replacing \( x \) with \(-x\) in any function generally reflects the function over the y-axis. This is because for every point \((x, y)\) on the function \( f(x) \), there is a corresponding point \((-x, y)\) on \( f(-x) \).

### Conclusion

Given that \( g(x) = 0.7 \cdot 6^{-x} \) is found by replacing \( x \) with \(-x\) in \( f(x) = 0.7 \cdot 6^x \), \( g(x) \) is indeed the reflection of \( f(x) \) over the y-axis.

Thus, the correct relationship between \( f(x) \) and \( g(x) \) is:
[tex]\[ g(x) \text{ is the reflection of } f(x) \text{ over the } y\text{-axis}. \][/tex]