Find f gx and gf x for each of the following pairs of functions. a) fx3x4; gx2x5
b) fxx2 5x; gx2x
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) = 6x + 8
b)
f[g(x)] = f[2x] = (2x)^2 + 5(2x) = 4x^2 + 10x
g[f(x)] = g[x^2 + 5x] = 2(x^2 + 5x) = 2x^2 + 10x
a) To find f[g(x)], we substitute g(x) into f(x):
f[g(x)] = 3(g(x)) + 4
Substituting g(x) = -2x + 5:
f[g(x)] = 3(-2x + 5) + 4
= -6x + 15 + 4
= -6x + 19
To find g[f(x)], we substitute f(x) into g(x):
g[f(x)] = -2(f(x)) + 5
Substituting f(x) = 3x + 4:
g[f(x)] = -2(3x + 4) + 5
= -6x - 8 + 5
= -6x - 3
Therefore, f[g(x)] = -6x + 19 and g[f(x)] = -6x - 3.
b) To find f[g(x)], we substitute g(x) into f(x):
f[g(x)] = (g(x))^2 + 5(g(x))
Substituting g(x) = 2x:
f[g(x)] = (2x)^2 + 5(2x)
= 4x^2 + 10x
To find g[f(x)], we substitute f(x) into g(x):
g[f(x)] = 2(f(x))
Substituting f(x) = x^2 + 5x:
g[f(x)] = 2(x^2 + 5x)
= 2x^2 + 10x
Therefore, f[g(x)] = 4x^2 + 10x and g[f(x)] = 2x^2 + 10x.