Let's analyze each pair of functions to determine which one satisfies the equation [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex].
### Pair 1: [tex]\( f(a) = a^2 - 4 \)[/tex] and [tex]\( g(a) = \sqrt{a} \)[/tex]
First, we determine [tex]\( g(f(a)) \)[/tex]:
[tex]\[ f(a) = a^2 - 4 \][/tex]
Then, apply [tex]\( g \)[/tex] to [tex]\( f(a) \)[/tex]:
[tex]\[ g(f(a)) = g(a^2 - 4) = \sqrt{a^2 - 4} \][/tex]
For [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex]:
[tex]\[ \sqrt{a^2 - 4} \stackrel{?}{=} |a| - 2 \][/tex]
This equation needs to be valid for all real values of [tex]\( a \)[/tex]. However, these expressions are not equivalent for all values of [tex]\( a \)[/tex]. Thus, this pair is not correct.
### Pair 2: [tex]\( f(a) = \frac{1}{2} a - 1 \)[/tex] and [tex]\( g(a) = 2a - 2 \)[/tex]
First, we determine [tex]\( g(f(a)) \)[/tex]:
[tex]\[ f(a) = \frac{1}{2} a - 1 \][/tex]
Then, apply [tex]\( g \)[/tex] to [tex]\( f(a) \)[/tex]:
[tex]\[ g(f(a)) = g\left(\frac{1}{2} a - 1\right) = 2\left(\frac{1}{2} a - 1\right) - 2 = a - 2 - 2 = a - 4 \][/tex]
For [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex]:
[tex]\[ a - 4 \stackrel{?}{=} |a| - 2 \][/tex]
This equation is not valid for all real values of [tex]\( a \)[/tex]. Thus, this pair is not correct.
### Pair 3: [tex]\( f(a) = 5 + a^2 \)[/tex] and [tex]\( g(a) = \sqrt{a - 5} - 2 \)[/tex]
First, we determine [tex]\( g(f(a)) \)[/tex]:
[tex]\[ f(a) = 5 + a^2 \][/tex]
Then, apply [tex]\( g \)[/tex] to [tex]\( f(a) \)[/tex]:
[tex]\[ g(f(a)) = g(5 + a^2) = \sqrt{5 + a^2 - 5} - 2 = \sqrt{a^2} - 2 = |a| - 2 \][/tex]
For [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex]:
[tex]\[ |a| - 2 \stackrel{?}{=} |a| - 2 \][/tex]
This equation is indeed valid for any real value of [tex]\( a \)[/tex]. Thus, this pair is correct.
### Pair 4: [tex]\( f(a) = 3 - 3a \)[/tex] and [tex]\( g(a) = 4a - 5 \)[/tex]
First, we determine [tex]\( g(f(a)) \)[/tex]:
[tex]\[ f(a) = 3 - 3a \][/tex]
Then, apply [tex]\( g \)[/tex] to [tex]\( f(a) \)[/tex]:
[tex]\[ g(f(a)) = g(3 - 3a) = 4(3 - 3a) - 5 = 12 - 12a - 5 = 7 - 12a \][/tex]
For [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex]:
[tex]\[ 7 - 12a \stackrel{?}{=} |a| - 2 \][/tex]
This equation is not valid for all real values of [tex]\( a \)[/tex]. Thus, this pair is not correct.
### Conclusion
The correct pair of functions for which [tex]\( (g \circ f)(a) = |a| - 2 \)[/tex] is:
[tex]\[ f(a) = 5 + a^2 \text{ and } g(a) = \sqrt{a - 5} - 2 \][/tex]
