ket f(x)=5 x − 4 and g(x) = x^2 − 8. Compute the following functions.
1. f(g(x)) =
2. g(f(x)) =
3. f(f(x)) =
4. g(g(x)) =
I will do one of them, then you do the others the same way ...
2.
g(f(x))
= g(5x-4)
= (5x-4)^2 - 8
To compute the given functions, we substitute the expression for one function into the other function wherever we see "x". Let's calculate each of these:
1. To find f(g(x)), we substitute g(x) into f(x):
f(g(x)) = f(x^2 - 8)
Now, replace x in f(x) with (x^2 - 8):
f(g(x)) = 5(x^2 - 8) - 4
Expanding the expression:
f(g(x)) = 5x^2 - 40 - 4
Combine like terms:
f(g(x)) = 5x^2 - 44
2. To find g(f(x)), we substitute f(x) into g(x):
g(f(x)) = g(5x - 4)
Now, replace x in g(x) with (5x - 4):
g(f(x)) = (5x - 4)^2 - 8
Expanding and simplifying:
g(f(x)) = 25x^2 - 40x + 16 - 8
Combine like terms:
g(f(x)) = 25x^2 - 40x + 8
3. To find f(f(x)), we substitute f(x) into itself:
f(f(x)) = f(5x - 4)
Now, replace x in f(x) with (5x - 4):
f(f(x)) = 5(5x - 4) - 4
Expanding and simplifying:
f(f(x)) = 25x - 20 - 4
Combine like terms:
f(f(x)) = 25x - 24
4. To find g(g(x)), we substitute g(x) into itself:
g(g(x)) = g(x^2 - 8)
Now, replace x in g(x) with (x^2 - 8):
g(g(x)) = (x^2 - 8)^2 - 8
Expanding and simplifying:
g(g(x)) = x^4 - 16x^2 + 64 - 8
Combine like terms:
g(g(x)) = x^4 - 16x^2 + 56
So the computed functions are:
1. f(g(x)) = 5x^2 - 44
2. g(f(x)) = 25x^2 - 40x + 8
3. f(f(x)) = 25x - 24
4. g(g(x)) = x^4 - 16x^2 + 56