To determine the new functional form of the given function [tex]\( f(x) = x^3 - 3 \)[/tex] after applying the specified transformations, we need to follow these steps:
1. Shift Right by 2 Units:
When a function is shifted horizontally to the right by 2 units, we replace [tex]\( x \)[/tex] with [tex]\( x - 2 \)[/tex] in the function. So, if the original function is [tex]\( f(x) = x^3 - 3 \)[/tex], the function after shifting right by 2 units will be:
[tex]\[
f(x - 2) = (x - 2)^3 - 3
\][/tex]
2. Shift Down by 3 Units:
When a function is shifted vertically downward by 3 units, we subtract 3 from the entire function. Applying this transformation to the function we obtained from the horizontal shift, we get:
[tex]\[
(x - 2)^3 - 3 - 3
\][/tex]
3. Simplify the Expression:
Combining both transformations, the new functional form becomes:
[tex]\[
(x - 2)^3 - 6
\][/tex]
Therefore, the new functional form of the function after being shifted right by 2 units and down by 3 units is:
[tex]\[
f(x) = (x - 2)^3 - 6
\][/tex]
