If f(x)=2x+5 and g(x)=8−3x^2, what is the value of f(g(4))−g(f(4))?

To find the value of f(g(4))−g(f(4)), we need to evaluate the functions f(x) and g(x) at x = 4, and then substitute the results into the expression.

Let's start by evaluating g(4) first.
g(x) = 8 - 3x^2
Substituting x = 4:
g(4) = 8 - 3(4)^2
g(4) = 8 - 3(16)
g(4) = 8 - 48
g(4) = -40

Now, let's evaluate f(4).
f(x) = 2x + 5
Substituting x = 4:
f(4) = 2(4) + 5
f(4) = 8 + 5
f(4) = 13

Now we have g(4) = -40 and f(4) = 13. We can substitute these values into the expression f(g(4))−g(f(4)):

f(g(4))−g(f(4)) = f(-40) - g(13)

Substituting into f(x):
f(-40) = 2(-40) + 5
f(-40) = -80 + 5
f(-40) = -75

Substituting into g(x):
g(13) = 8 - 3(13)^2
g(13) = 8 - 3(169)
g(13) = 8 - 507
g(13) = -499

Now we have f(-40) = -75 and g(13) = -499. Substituting these values into the expression:
f(g(4)) - g(f(4)) = (-75) - (-499)
f(g(4)) - g(f(4)) = -75 + 499
f(g(4)) - g(f(4)) = 424

Therefore, the value of f(g(4))−g(f(4)) is 424.