To determine the equation of the new function when the linear parent function [tex]\( f(x) = x \)[/tex] is shifted down by 7 units, follow these steps:
1. Identify the linear parent function:
The linear parent function is [tex]\( f(x) = x \)[/tex].
2. Understand the effect of the transformation:
Shifting a function down by a certain number of units involves subtracting that number from the function's output. In this case, we want to shift the function down by 7 units.
3. Apply the transformation:
For every [tex]\( x \)[/tex] value, [tex]\( f(x) = x \)[/tex] will be decreased by 7 units. Mathematically, this is represented as:
[tex]\[
g(x) = f(x) - 7
\][/tex]
Since [tex]\( f(x) = x \)[/tex], substituting [tex]\( f(x) \)[/tex] into the equation gives:
[tex]\[
g(x) = x - 7
\][/tex]
4. Write the new function:
So, the equation of the new function after shifting [tex]\( f(x) = x \)[/tex] down by 7 units is:
[tex]\[
g(x) = x - 7
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{g(x) = x - 7}
\][/tex]
Therefore, the correct option is D: [tex]\( g(x) = x - 7 \)[/tex].
