To find the value of [tex]\( f(g(3)) \)[/tex], we first need to determine what [tex]\( g(3) \)[/tex] equals and then use that result in the function [tex]\( f(x) \)[/tex]. Let's do this step by step.
1. Evaluate [tex]\( g(3) \)[/tex]:
[tex]\[ g(x) = \frac{x + 1}{x - 1} \][/tex]
Substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(3) = \frac{3 + 1}{3 - 1} = \frac{4}{2} = 2.0 \][/tex]
So, [tex]\( g(3) = 2.0 \)[/tex].
2. Evaluate [tex]\( f(g(3)) \)[/tex] or [tex]\( f(2.0) \)[/tex]:
[tex]\[ f(x) = 3x - 7 \][/tex]
Substitute [tex]\( x = 2.0 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(2.0) = 3(2.0) - 7 = 6.0 - 7 = -1.0 \][/tex]
So, [tex]\( f(2.0) = -1.0 \)[/tex].
Thus, the value of [tex]\( f(g(3)) \)[/tex] is [tex]\(-1.0\)[/tex].
In summary, the steps are:
1. Compute [tex]\( g(3) \)[/tex] to obtain [tex]\( 2.0 \)[/tex].
2. Use this result in [tex]\( f(x) \)[/tex] to find [tex]\( f(2.0) = -1.0 \)[/tex].
Therefore, [tex]\( f(g(3)) = -1.0 \)[/tex].
