Answer:
[tex](g-h)(x) = 3x+11[/tex]
[tex](g+h)(x) = 5x+1[/tex]
[tex](g-h)(3) =20[/tex]
[tex](g+h)(3) =16[/tex]
Step-by-step explanation:
Given functions:
[tex]\begin{cases}g(x)=4x+6\\h(x)=x-5\end{cases}[/tex]
Function composition is an operation that takes two functions and produces a third function.
Therefore, the composite function (g-h)(x) means subtract function h(x) from function g(x). Similarly, (g+h)(x) means to add function h(x) to function g(x).
[tex]\begin{aligned}(g-h)(x) & = g(x)-h(x)\\& = (4x+6)-(x-5)\\& = 4x+6-x+5\\& = 4x-x+6+5\\& = 3x+11\end{aligned}[/tex]
[tex]\begin{aligned}(g+h)(x) & = g(x)+h(x)\\& = (4x+6)+(x-5)\\& = 4x+6+x-5\\& = 4x+x+6-5\\& = 5x+1\end{aligned}[/tex]
To evaluate both composite functions when x = 3, simply substitute x = 3 into the found composite functions:
[tex]\begin{aligned}(g-h)(3) & = 3(3)+11\\& = 9+11\\&=20\end{aligned}[/tex]
[tex]\begin{aligned}(g+h)(3) & = 5(3)+1\\& = 15+11\\&=16\end{aligned}[/tex]
