To determine the function [tex]\( g(x) \)[/tex] when the graph of the absolute value parent function, [tex]\( f(x) = |x| \)[/tex], is stretched horizontally by a factor of 6, follow these steps:
1. Understand Horizontal Stretch: When a function is stretched horizontally by a factor, it affects the input variable [tex]\( x \)[/tex]. Specifically, to stretch a function [tex]\( f(x) \)[/tex] horizontally by a factor of [tex]\( k \)[/tex], we replace [tex]\( x \)[/tex] with [tex]\( \frac{x}{k} \)[/tex] in the function.
2. Apply the Horizontal Stretch:
- Here, we need a horizontal stretch by a factor of [tex]\( 6 \)[/tex].
- This means in the function [tex]\( f(x) = |x| \)[/tex], we will replace [tex]\( x \)[/tex] with [tex]\( \frac{x}{6} \)[/tex].
3. Substitute and Simplify:
- The modified function after applying the horizontal stretch is [tex]\( g(x) = \left| \frac{x}{6} \right| \)[/tex].
4. Choose the Correct Option: We now compare our result with the given options:
- A. [tex]\( g(x) = |x + 6| \)[/tex]
- B. [tex]\( g(x) = 6|x| \)[/tex]
- C. [tex]\( g(x) = |6x| \)[/tex]
- D. [tex]\( g(x) = \left| \frac{1}{6} x \right| \)[/tex]
The correct choice that matches our derived function [tex]\( \left| \frac{x}{6} \right| \)[/tex] is option D: [tex]\( g(x) = \left| \frac{1}{6} x \right| \)[/tex].
So, the function [tex]\( g(x) \)[/tex] is indeed:
[tex]\[ g(x) = \left| \frac{1}{6} x \right| \][/tex]
Thus, the correct answer is option D.
