Find
f gx
and
gf x
for each of the following pairs of functions.
a)
f x 3x 4
;
gx 2x 5
b)
f x x 5x
2
;
gx 2x 1
a)
f[g(x)] = f(-2x + 5) = 3(-2x + 5) + 4 = -6x + 15 + 4 = -6x + 19
g[f(x)] = g(3x + 4) = -2(3x + 4) + 5 = -6x - 8 + 5 = -6x - 3
b)
f[g(x)] = f(2x + 1) = (2x + 1)^2 + 5(2x + 1) = 4x^2 + 4x + 1 + 10x + 5 = 4x^2 + 14x + 6
g[f(x)] = g(x^2 + 5x) = 2(x^2 + 5x) + 1 = 2x^2 + 10x + 1
a) To find f[g(x)], substitute g(x) into the function f(x):
f[g(x)] = 3(g(x)) + 4
Now substitute the expression for g(x):
f[g(x)] = 3(-2x + 5) + 4
Simplify the expression:
f[g(x)] = -6x + 15 + 4
Combine like terms:
f[g(x)] = -6x + 19
To find g[f(x)], substitute f(x) into the function g(x):
g[f(x)] = -2(f(x)) + 5
Now substitute the expression for f(x):
g[f(x)] = -2(3x + 4) + 5
Simplify the expression:
g[f(x)] = -6x - 8 + 5
Combine like terms:
g[f(x)] = -6x - 3
b) To find f[g(x)], substitute g(x) into the function f(x):
f[g(x)] = (g(x))^5 + 5(g(x))^2
Now substitute the expression for g(x):
f[g(x)] = (2x + 1)^5 + 5(2x + 1)^2
To simplify this expression further, we would need to expand the polynomial and simplify.
To find g[f(x)], substitute f(x) into the function g(x):
g[f(x)] = 2(f(x)) + 1
Now substitute the expression for f(x):
g[f(x)] = 2((x^2 + 5x) + 1) + 1
Simplify the expression:
g[f(x)] = 2x^2 + 10x + 2 + 1
Combine like terms:
g[f(x)] = 2x^2 + 10x + 3