Question:
Solution:
Consider the following functions:
[tex]f(x)\text{ = x+4}[/tex][tex]g(x)\text{ = }4x-3[/tex]then:
[tex](f\circ g)(x)\text{ = f(g(x))= (4x-3)+4}[/tex]this is equivalent to:
[tex](f\circ g)(x)\text{ =(4x-3)+4 = 4x +1}[/tex]thus:
[tex](f\circ g)(x)\text{ =4x +1}[/tex]So, replacing x = 2 in the previous function we get:
[tex](f\circ g)(2)\text{ =4(2) +1 = 9}[/tex]then:
[tex](f\circ g)(2)\text{ = 9}[/tex]On the other hand, the composition:
[tex](g\circ f)(x)\text{ = g(f(x)) = 4(x+4)-3}[/tex]this is equivalent to:
[tex](g\circ f)(x)\text{ = 4(x+4)-3 = 4x +16-3 = 4x - 13}[/tex]then, we can conclude that:
[tex](g\circ f)(x)\text{ =4x - 13}[/tex]So, replacing x = 2 in the previous function we get:
[tex](g\circ f)(2)\text{ =4(2) - 13 = 8-13 = -5}[/tex]then:
[tex](g\circ f)(2)\text{ =4(2) - 13 = 8-13 = -5}[/tex]then:
[tex](g\circ f)(2)\text{ =-5}[/tex]Then the correct answers are:
1)
[tex](f\circ g)(x)\text{ =4x +1}[/tex]2)
[tex](f\circ g)(2)\text{ = 9}[/tex]3)
[tex](g\circ f)(x)\text{ =4x - 13}[/tex]4)
[tex](g\circ f)(2)\text{ =-5}[/tex]
