3) Given
3 1
2
f x x , find
gf x
and
f gx
using each function
gx
below.
a)
2
g x x x 2 3
b)
; 0
1
x
x
To find g(f(x)), we substitute f(x) into g(x):
a) g(f(x)) = g(x - 3) = -(x - 3)^2/3
b) g(f(x)) = g(x - 1) = (x - 1)^2
To find f(g(x)), we substitute g(x) into f(x):
a) f(g(x)) = f(-(2x^3)) = -(2x^3 - 3) = -2x^3 + 3
b) f(g(x)) = f(x - 1) = (x - 1 - 3)^2/3 = (x - 4)^2/3
To find g(f(x)) and f(g(x)) using each function g(x) given, you need to substitute f(x) or g(x) into the respective functions. Let's go through each case.
a) Given g(x) = x^2 - 2x^3:
To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = (f(x))^2 - 2(f(x))^3
Substituting f(x) = x - 3:
g(f(x)) = (x - 3)^2 - 2(x - 3)^3
= (x - 3)(x - 3) - 2(x - 3)(x - 3)(x - 3)
= (x^2 - 6x + 9) - 2(x^3 - 9x^2 + 27x - 27)
= x^2 - 6x + 9 - 2x^3 + 18x^2 - 54x + 54
= -2x^3 + 18x^2 - 60x + 63
To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = g(x) - 3
Substituting g(x) = x^2 - 2x^3:
f(g(x)) = (x^2 - 2x^3) - 3
= x^2 - 2x^3 - 3
= -2x^3 + x^2 - 3
So, g(f(x)) = -2x^3 + 18x^2 - 60x + 63 and f(g(x)) = -2x^3 + x^2 - 3.
b) Given g(x) = x^0.5; x ≠ 0:
To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = (f(x))^0.5
Substituting f(x) = x - 3:
g(f(x)) = (x - 3)^0.5
To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = g(x) - 3
Substituting g(x) = x^0.5; x ≠ 0:
f(g(x)) = (x^0.5) - 3
= √x - 3
So, g(f(x)) = (x - 3)^0.5 and f(g(x)) = √x - 3.
Note: In case b), f(g(x)) is only defined for x > 0 as taking the square root of a negative number would result in an imaginary number.