f(x) 2x2 + 3 and g(x) = 4x + 5 find and simplify
a) (g ◦ f)(x) =
b) (f ◦ g )(x)=
c) (f ◦ g )(2) =
I will do b) which will allow you to do c)
b) (f ◦ g )(x)= = f(g(x) )
= f(4x+5)
= 2(4x+5)^2 + 3
for c) replace x with 2 and evaluate
for a) follow my steps I used for b)
f(x) = 2 x^2 + 3 and g(cow) = 4cow + 5
g [f (x)] = 4 f(x)+5
= 4 *(2x^2+3) + 5
= 8 x^2 + 12 + 5
= 8 x^2 +17
f[ g(x)] = 2 (g^2) +3
= 2[ 16x^2 + 40 x +25] + 3
= 32 x^2 + 80 x + 53
32*4 + 160 + 53
=128 + 160 + 53
=
To find and simplify the compositions of functions, we need to substitute the inner function into the outer function.
a) To find (g ◦ f)(x), we substitute f(x) into g(x):
(g ◦ f)(x) = g[f(x)] = g[2x^2 + 3]
Now we substitute the expression 2x^2 + 3 for x in g(x):
g[2x^2 + 3] = 4(2x^2 + 3) + 5
Simplifying further:
= 8x^2 + 12 + 5
= 8x^2 + 17
Therefore, (g ◦ f)(x) simplifies to 8x^2 + 17.
b) To find (f ◦ g)(x), we substitute g(x) into f(x):
(f ◦ g)(x) = f[g(x)] = f[4x + 5]
Now we substitute the expression 4x + 5 for x in f(x):
f[4x + 5] = 2(4x + 5)^2 + 3
Expanding and simplifying:
= 2(16x^2 + 40x + 25) + 3
= 32x^2 + 80x + 50 + 3
= 32x^2 + 80x + 53
Therefore, (f ◦ g)(x) simplifies to 32x^2 + 80x + 53.
c) To find (f ◦ g)(2), we substitute 2 into g(x) and then substitute the result into f(x):
(g(2) = 4(2) + 5 = 8 + 5 = 13
(f ◦ g)(2) = f[g(2)] = f(13)
Now we substitute 13 into f(x):
f(13) = 2(13)^2 + 3
= 2(169) + 3
= 338 + 3
= 341
Therefore, (f ◦ g)(2) simplifies to 341.