Question

Question Given the function f(x)=log8x , construct the function g(x) such that the graph of g(x) is the same as the graph of f(x) but shifted 7 units down, shrunk vertically by a factor of 15 , and reflected across the y-axis. In 1–2 sentences, describe the construction of the function and identify the correct function.(2 points)

Answer 1

Answer:

$g(x)=\dfrac{1}{15}\log_8(-x)-\dfrac{7}{15}$

Step-by-step explanation:

Given logarithmic function:

$f(x)=\log_8(x)$

We are told that the graph of g(x) is the same as the graph of f(x) but shifted 7 units down, shrunk vertically by a factor of 15, and reflected across the y-axis.

To construct the function g(x), we should follow the order of the given transformations.

First, shift the function 7 units down by subtracting 7 from the function:

$f(x)-7=\log_8(x)-7$

Next, to vertically shrink the function by a factor of 15, we multiply the function by 1/15:

$\dfrac{1}{15}\left(f(x)-7\right)=\dfrac{1}{15}\left(\log_8(x)-7\right)$

Finally, to reflect the function across the y-axis, we negate the x-values:

$\dfrac{1}{15}\left(f(-x)-7\right)=\dfrac{1}{15}\left(\log_8(-x)-7\right)$

Therefore, function g(x) is:

$g(x)=\dfrac{1}{15}\left(\log_8(-x)-7\right)$

This simplifies to:

$\Large\boxed{\boxed{g(x)=\dfrac{1}{15}\log_8(-x)-\dfrac{7}{15}}}$