Question

For f(x) = x^2+6 and g(x) = x^2-5 find the following functions.

a.) (f o g)(x)

b.) (g o f) (x)

c.) (f o g) (4)

d.) (g o f) (4)

Answer 1

f(x) = x² + 6

g(x) = x² - 5

a) (f o g)(x) = f(g(x))

f(g(x)) = f(x² - 5) = (x² - 5)² + 6

f(g(x)) = x⁴ - 10x² + 25 + 6

f(g(x)) = x⁴ - 10x² + 31

b) (g o f)(x) = g(f(x))

g(f(x)) = g(x² + 6) = (x² + 6)² - 5

g(f(x)) = x⁴ + 12x² + 36 - 5

g(f(x)) = x⁴ + 12x² + 31

c) (f o g)(4) = f(g(4))

f(g(x)) = x⁴ - 10x² + 31

Putting x = 4

f(g(4)) = (4)⁴ - 10(4)² + 31 = 256 - 160 + 31 = 127

f(g(4)) = 127

d) (g o f)(4) = g(f(4))

g(f(x)) = x⁴ + 12x² + 31

g(f(4)) = (4)² + 12(4)² + 31 = 256 + 192 + 31 = 479

g(f(4)) = 479