Sure, let's break down the transformations step by step to understand how the new function is derived from the parent function [tex]\( f(x) = |x| \)[/tex].
1. Shift 5 units to the right:
- When we shift a function horizontally, we replace [tex]\( x \)[/tex] with [tex]\( x - h \)[/tex] where [tex]\( h \)[/tex] is the number of units we're shifting. In this case, we're shifting 5 units to the right, so we replace [tex]\( x \)[/tex] with [tex]\( x - 5 \)[/tex].
- Therefore, [tex]\( f(x) = |x| \)[/tex] becomes [tex]\( f(x) = |x - 5| \)[/tex].
2. Shift 7 units down:
- When we shift a function vertically, we subtract [tex]\( k \)[/tex] from the function where [tex]\( k \)[/tex] is the number of units we're shifting. In this case, we're shifting 7 units down, so we subtract 7 from the function.
- Therefore, [tex]\( f(x) = |x - 5| \)[/tex] becomes [tex]\( g(x) = |x - 5| - 7 \)[/tex].
Putting these steps together, the new function [tex]\( g(x) \)[/tex] after applying both transformations is:
[tex]\[ g(x) = |x - 5| - 7 \][/tex]
Therefore, the correct answer is:
D. [tex]\( g(x) = |x - 5| - 7 \)[/tex]
