Question

Given the function f(x)=log5 x which of the following functions is the transformation of f(x) right 6 units up 3 units stretched vertically by a factor of 2 and reflected across the x axis
All my answers start with g(x) and end with either plus 3 or minus 3

Answer 1

To determine the transformation of the function \( f(x) = \log_5 x \) based on the given conditions (shift right 6 units, shift up 3 units, stretched vertically by a factor of 2, and reflected across the x-axis), let's apply each transformation step by step.

1. **Shift right 6 units**:
- This means we replace \( x \) with \( x - 6 \).
- So, \( f(x) \to f(x - 6) = \log_5(x - 6) \).

2. **Stretch vertically by a factor of 2**:
- This means we multiply the output of the function by 2.
- So, \( f(x - 6) \to 2f(x - 6) = 2 \log_5(x - 6) \).

3. **Reflect across the x-axis**:
- This means we multiply the entire function by -1.
- So, \( 2f(x - 6) \to -2f(x - 6) = -2 \log_5(x - 6) \).

4. **Shift up 3 units**:
- This means we add 3 to the entire function.
- So, \( -2f(x - 6) \to -2\log_5(x - 6) + 3 \).

Putting it all together, the transformed function is:
\[
g(x) = -2 \log_5(x - 6) + 3
\]

So, \( g(x) \) will end with "+ 3".