To solve the given problem step-by-step, follow these procedures:
1. Given function:
[tex]\[
g(n) = n^2 - 5n^2
\][/tex]
2. Simplify the expression for [tex]\( g(n) \)[/tex]:
[tex]\[
g(n) = n^2 - 5n^2 = -4n^2
\][/tex]
3. Substitute [tex]\( n \)[/tex] with [tex]\((n' = -4n + 1)\)[/tex] in the simplified function [tex]\( g(n) = -4n^2 \)[/tex], meaning, we need to find [tex]\( g(-4n + 1) \)[/tex]:
Let's denote [tex]\( n' = -4n + 1 \)[/tex], and substitute [tex]\( n' \)[/tex] into the expression:
[tex]\[
g(n') = -4(n')^2
\][/tex]
4. Substitute the value of [tex]\( n' = -4n + 1 \)[/tex] into [tex]\( g(n') \)[/tex]:
[tex]\[
g(-4n + 1) = -4((-4n + 1))^2
\][/tex]
5. Simplify the expression:
Expanding the square:
[tex]\[
(-4n + 1)^2 = (1 - 4n)^2 = (1 - 4n)(1 - 4n) = 1 - 8n + 16n^2
\][/tex]
Substituting back into [tex]\( g(-4n + 1) \)[/tex]:
[tex]\[
g(-4n + 1) = -4(1 - 8n + 16n^2) = -4 + 32n - 64n^2
\][/tex]
Therefore, the simplified functions for [tex]\( g \)[/tex] and [tex]\( g \)[/tex] when substituting [tex]\( n \)[/tex] with [tex]\((-4n + 1)\)[/tex] are:
[tex]\[
g(n) = -4n^2
\][/tex]
and
[tex]\[
g(-4n + 1) = -4(1 - 4n)^2
\][/tex]
Expressing this result in the required format:
[tex]\[
g(n) = -4 n^{\wedge} 2
\][/tex]
and
[tex]\[
g(-4n + 1) = -4 (1 - 4 n)^{\wedge} 2
\][/tex]
