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Given that [tex]$f(x)=4x+2$[/tex] and [tex]$g(x)=x^2-2x-5$[/tex], find [tex][tex]$(g \circ f)(3)$[/tex][/tex].

Sagot :

To solve for [tex]\((g \circ f)(3)\)[/tex], we need to follow these steps:

1. Find [tex]\( f(3) \)[/tex]:
- We are given the function [tex]\( f(x) = 4x + 2 \)[/tex].
- Substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(3) = 4(3) + 2 = 12 + 2 = 14 \][/tex]

2. Now, use the result of [tex]\( f(3) \)[/tex] to find [tex]\( g(f(3)) \)[/tex]:
- We found [tex]\( f(3) = 14 \)[/tex].
- Next, we are given the function [tex]\( g(x) = x^2 - 2x - 5 \)[/tex].
- Substitute [tex]\( x = 14 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(14) = 14^2 - 2(14) - 5 \][/tex]

3. Calculate [tex]\( g(14) \)[/tex]:
- First, compute [tex]\( 14^2 \)[/tex]:
[tex]\[ 14^2 = 196 \][/tex]
- Then, compute [tex]\( 2(14) \)[/tex]:
[tex]\[ 2(14) = 28 \][/tex]
- Now, substitute these values back into [tex]\( g(14) \)[/tex]:
[tex]\[ g(14) = 196 - 28 - 5 = 168 - 5 = 163 \][/tex]

Therefore, the value of [tex]\((g \circ f)(3)\)[/tex] is [tex]\(163\)[/tex].