To determine the function [tex]\( g(x) \)[/tex] from the given transformation, let's carefully analyze the stretch being applied to the parent function [tex]\( f(x) = |x| \)[/tex].
When we apply a horizontal stretch by a factor of 6 to a function, each point [tex]\( (x, y) \)[/tex] on the graph of the original function moves to [tex]\( (6x, y) \)[/tex]. To achieve a horizontal stretch by a factor of 6 mathematically, we need to divide the [tex]\( x \)[/tex]-values by 6 before taking the absolute value. This means applying a factor of [tex]\( \frac{1}{6} \)[/tex] inside the absolute value function.
So, starting with:
[tex]\[ f(x) = |x| \][/tex]
For a horizontal stretch by a factor of 6:
[tex]\[ g(x) = f\left(\frac{x}{6}\right) \][/tex]
[tex]\[ g(x) = \left|\frac{x}{6}\right| \][/tex]
We can simplify [tex]\(\left|\frac{x}{6}\right|\)[/tex] to:
[tex]\[ g(x) = \left|\frac{1}{6} x\right| \][/tex]
Given the options, this matches with option C:
[tex]\[ \text{C. } g(x) = \left|\frac{1}{6} x\right| \][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] after applying the horizontal stretch by a factor of 6 is:
[tex]\[ g(x) = \left|\frac{1}{6} x\right| \][/tex]
So, the correct choice is:
C. [tex]\( g(x) = \left|\frac{1}{6} x\right| \)[/tex]
Thus, the answer is 3.
