Question

Can you help me it’s. 2 part question I can only add one picture at a time

Answer 1

Recall the definition of the composition between two functions:

$(f\circ g)(x)=f(g(x))$

Similarly, but not always the same:

$(g\circ f)(x)=g(f(x))$

Use the rules of correspondence of the functions f and g to find the compositions:

$\begin{gathered} f(x)=6x-10 \\ g(x)=8-4x \end{gathered}$

Then:

$\begin{gathered} (f\circ g)(x)=f(g(x)) \\ =6\cdot g(x)-10 \\ =6(8-4x)-10 \\ =6\cdot8-6\cdot4x-10 \\ =48-24x-10 \\ =-24x+38 \\ \\ \therefore(f\circ g)(x)=-24x+38 \end{gathered}$

On the other hand:

$\begin{gathered} (g\circ f)(x)=g(f(x)) \\ =8-4\cdot f(x) \\ =8-4(6x-10) \\ =8-4\cdot6x-4\cdot-10 \\ =8-24x+40 \\ =-24x+48 \\ \\ \therefore(g\circ f)(x)=-24x+48 \end{gathered}$

Therefore, the answers are:

$\begin{gathered} (f\circ g)(x)=-24x+38 \\ (g\circ f)(x)=-24x+48 \end{gathered}$