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]
### 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]
