Hi there!
[tex]f(x)+f(x+1)=4f(x)[/tex]
There are 3 parts to this equation:
f(x)
f(x+1)
4f(x)
We must first determine these three parts separately.
1) f(x)
We're given that [tex]f(x)=3^x[/tex]:
⇒ [tex]f(x)=3^x[/tex]:
2) f(x+1)
Now, we must find f(x+1). To do so, add 1 to x in the original function [tex]f(x)=3^x[/tex]:
⇒ [tex]f(x+1)=3^x^+^1[/tex]
3) 4f(x)
To find 4f(x), multiply the original function [tex]f(x)=3^x[/tex] by 4:
[tex]4f(x)=4*3^x[/tex]:
4) Put it all together
Now, plug each of the three parts into the equation [tex]f(x)+f(x+1)=4f(x)[/tex]:
[tex]f(x)+f(x+1)=4f(x)[/tex]
[tex]3^x+3^x^+^1=4*3^x\\3^x+3^x*3=4*3^x[/tex]
Factor the left side
[tex]3^x*(1+3)=4*3^x[/tex]
Divide both sides by 3^x
[tex]1+3=4\\4=4[/tex]
Because this equation is true, [tex]f(x)+f(x+1)=4f(x)[/tex] is therefore true.
I hope this helps!
