To determine the pair of functions [tex]\( f(a) \)[/tex] and [tex]\( g(x) \)[/tex] such that the composition [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex], we can break down the problem step-by-step.
1. Assume a function [tex]\( f(a) \)[/tex]: Let's consider the function [tex]\( f(a) \)[/tex] to be the absolute value function, i.e., [tex]\( f(a) = |a| \)[/tex].
2. Define [tex]\( g(x) \)[/tex]: We need to find [tex]\( g(x) \)[/tex] such that when we apply [tex]\( g \)[/tex] to [tex]\( f(a) \)[/tex], the result is [tex]\( |a| - 2 \)[/tex]. Since [tex]\( f(a) \)[/tex] is equal to [tex]\( |a| \)[/tex], we need [tex]\( g \)[/tex] to reduce [tex]\( |a| \)[/tex] by 2. Therefore, we can define [tex]\( g(x) \)[/tex] as [tex]\( g(x) = x - 2 \)[/tex].
To verify, let's see how these functions work together.
- For a given [tex]\( a \)[/tex], first apply [tex]\( f \)[/tex]:
[tex]\[
f(a) = |a|
\][/tex]
- Then apply [tex]\( g \)[/tex]:
[tex]\[
g(f(a)) = g(|a|) = |a| - 2
\][/tex]
Thus, the pair of functions [tex]\( f(a) = |a| \)[/tex] and [tex]\( g(x) = x - 2 \)[/tex] indeed satisfies the condition [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex].
Let's check this with an example:
- Let [tex]\( a = -5 \)[/tex]:
- [tex]\( f(a) = |-5| = 5 \)[/tex]
- [tex]\( g(f(a)) = g(5) = 5 - 2 = 3 \)[/tex]
The result matches [tex]\( |a| - 2 \)[/tex] because [tex]\( |-5| - 2 = 5 - 2 = 3 \)[/tex].
Thus, the functions are:
[tex]\[
f(a) = |a|
\][/tex]
[tex]\[
g(x) = x - 2
\][/tex]
