f(x)=x^2-25 g(x)=x+4solve fg(x+1)=gf(x)

First, let's find the expressions for fg(x) and gf(x).

fg(x) is the composition of f and g, which means we apply g first and then f. So, fg(x) = f(g(x)).

Given g(x) = x + 4, then g(x+1) = (x+1) + 4 = x + 5.

So, fg(x+1) = f(g(x+1)) = f(x+5) = (x+5)^2 - 25.

Similarly, gf(x) is the composition of g and f, which means we apply f first and then g. So, gf(x) = g(f(x)).

Given f(x) = x^2 - 25, then f(x) = x^2 - 25.

So, gf(x) = g(f(x)) = g(x^2 - 25) = (x^2 - 25) + 4 = x^2 - 21.

The equation fg(x+1) = gf(x) becomes (x+5)^2 - 25 = x^2 - 21.

Solving this equation, we get:

(x+5)^2 - x^2 = 4 x^2 + 10x + 25 - x^2 = 4 10x + 25 = 4 10x = -21 x = -21/10 = -2.1

So, the solution to the equation fg(x+1) = gf(x) is x = -2.1.