The transformation that shifts a function along the [tex]x-axis[/tex] is [tex]f(x+a)[/tex], which shifts the function to the left, and [tex]f(x-a)[/tex], which shifts the function to the right.
The transformation that shifts a function along the [tex]y-axis[/tex] is [tex]f(x)+a[/tex], which shifts the function up, and [tex]f(x)-a[/tex], which shifts the function down
Hence, the correct answer is D: [tex]g(x)=f(x+8)-3[/tex]