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

Answer 1

To transform the function \( f(x) = \log_5 x \) according to the specified transformations, we'll proceed step by step:

1. **Right 6 units**: This transformation is done by replacing \( x \) with \( x - 6 \). So, we have:
\[
f_1(x) = \log_5(x - 6)
\]

2. **Up 3 units**: This involves adding 3 to the entire function:
\[
f_2(x) = \log_5(x - 6) + 3
\]

3. **Stretched vertically by a factor of 2**: This means we multiply the function by 2:
\[
f_3(x) = 2(\log_5(x - 6) + 3) = 2\log_5(x - 6) + 6
\]

4. **Reflected across the x-axis**: This changes the sign of the entire function:
\[
f(x) = -[2\log_5(x - 6) + 6] = -2\log_5(x - 6) - 6
\]

Putting all the transformations together, the final transformed function is:
\[
f(x) = -2\log_5(x - 6) - 6
\]

This represents the function \( f(x) = \log_5 x \) after the specified transformations.