To determine the equation of the new function resulting from the given transformations to the linear parent function [tex]\( f(x) = x \)[/tex], we will follow these steps:
Step 1: Vertically stretch by a factor of 4
A vertical stretch of a function by a factor [tex]\( k \)[/tex] involves multiplying the function by [tex]\( k \)[/tex].
For the linear parent function [tex]\( f(x) = x \)[/tex], stretching it vertically by a factor of 4 results in:
[tex]\[ f(x) = 4x \][/tex]
Step 2: Flip over the [tex]\( x \)[/tex]-axis
Flipping a function over the [tex]\( x \)[/tex]-axis involves multiplying the function by [tex]\(-1\)[/tex].
Applying this transformation to our equation from Step 1, [tex]\( f(x) = 4x \)[/tex]:
[tex]\[ f(x) = -4x \][/tex]
Therefore, the correct equation of the new function after applying the given transformations is [tex]\( g(x) = -4x \)[/tex].
Hence, the correct answer is:
D. [tex]\( g(x) = -4x \)[/tex]
