The graph is a plot of an absolute value function.
The general equation of an absolute value function is expressed as
[tex]\begin{gathered} y=a\lvert x-h\rvert+k \\ \text{where} \\ h=x-\text{coordinate of the vertex of the plot} \\ k=\text{ y-coordinate of the vertex of the plot} \\ a\text{ = constant} \end{gathered}[/tex]
Thus, given the graph below
[tex]\begin{gathered} \text{the values of h and k at the vertex of the plot is evaluated as} \\ (h,k)=(0,\text{ -4)} \end{gathered}[/tex]
Thus,
[tex]\begin{gathered} y=a\lvert x-0\rvert+(-4)\text{ } \\ \Rightarrow\text{ y= a}\lvert x\rvert-4\text{ ------ equation 1} \end{gathered}[/tex]
From the graph, taking either zeros of the function (that is, x = -4 or x = 4)
[tex]\begin{gathered} \text{when x = -4, y = 0} \\ substitute\text{ the above values in equation 1} \\ \Rightarrow0=a\lvert-4\rvert-4 \\ 0=4a-4 \\ collect\text{ like terms} \\ 4a=0+4 \\ 4a=4 \\ a=\frac{4}{4}=1 \\ a=1 \end{gathered}[/tex]
Thus, the equation of the plot is given as
[tex]\begin{gathered} f(x)=1\lvert x-0\rvert-4 \\ \Rightarrow f(x)\text{ = }\lvert x\rvert-4 \end{gathered}[/tex]
To solve for g(x):
[tex]\begin{gathered} g(x)=f(4x) \\ \Rightarrow f(4x)=\lvert4x\rvert-4 \end{gathered}[/tex]
Thus,
[tex]g(x)\text{ = }\lvert4x\rvert-4[/tex]
