f(x) = 2x+3 g(x) = -x^3 How could you find the following Show fully worked solutions please 1. g(g(x)) 2. f(f(x)) 3. f(f+g)(x) 4. f(f-g)(x)?

Question

5. f(fg)(x)

Thanks

Answer 1

f(x) = 2x + 3, g(x) = -x^3



[ 1 ].

g(g(x)) = -(-x^3)^3 = x^9



[ 2 ].

f(f(x)) = 2(2x + 3) + 3

= 4x + 6 + 3

= 4x + 9



[ 3 ].

f(f + g)(x) = 2(2x + 3 -x^3) + 3

= 4x + 6 -2x^3 + 3

= -2x^3 + 4x + 9



[ 4 ].

f(f - g)(x) = 2(2x + 3 -(-x^3)) + 3

= 4x + 6 + 2x^3 + 3

= 2x^3 + 4x + 9



[ 5 ].

f(fg)(x) = f((2x+3)*-x^3)(x)

= f(-2x^4 -3x^3)(x)

= 2(-2x^4 -3x^3) + 3

= -4x^4 -6x^3 + 3

Answer 2

f(x) = 2x+3

g(x) = -x^3



1)

g (g(x)) = g(-x^3) = -(-x^3)^3 = -(-x^9) = x^9



2)

f(f(x)) = f(2x+3) = 2(2x+3)+3 = 4x+6+3 = 4x+9



3)

f((f+g))(x)

(f+g)(x) = f(x)+g(x) = 2x+3-x^3

f ( (f+g)(x)) = f( -x^3+2x+3) = 2(-x^3+2x+3)+ 3 = -2x^3+4x+6+3 = -2x^3+4x+9



4)

(f-g)(x) = f(x)-g(x) = 2x+3 -(-x^3) = x^3+2x+3

f((f-g)(x)) = f(x^3+2x+3) = 2(x^3+2x+3)+3 = 2x^3+4x+9



5)

fg(x) = f(x)*g(x) = (2x+3)(-x^3) = -2x^4-3x^3

f( fg(x) ) = f(-2x^4-3x^3) = 2(-2x^4-3x^3)+3 = -4x^4-6x^3+3

Answer 3

1)

g(g(x)) =

-(-x^3)^3 =

-(-x^9) =

x^9



One question at a time.

Answer 4

　

f(x) = 2x + 3

g(x) = −x³



1. g(g(x)) = −(g(x))³ = −(−x³)³ = x⁹

2. f(f(x)) = 2f(x) + 3 = 2(2x+3) + 3 = 4x + 9

3. f(f+g)(x) = 2(f(x)+g(x)) + 3 = 2(2x+3−x³) + 3 = −2x³ + 4x + 9

4. f(f−g)(x) = 2(f(x)−g(x)) + 3 = 2(2x+3−(−x³)) + 3 = 2x³ + 4x + 9

5. f(fg)(x) = 2(f(x)*g(x)) + 3 = 2(2x+3)(−x³) + 3 = −4x⁴ − 6x³ + 3

Answer 5

1. g(g(x)) = g(-x³) = -(-x³)³ = x⁹

2. f(f(x)) = f(2x+3) = 2(2x+3)+3 = 4x+9

3. f((f+g)(x)) = f(f(x)+g(x)) = f(2x+3 + (-x³)) = f(-x³+2x+3) = 2(-x³+2x+3)+3 = -2x³+4x+9

4. f(f-g(x)) = f(f(x)-g(x)) = f(2x+3 - (-x³)) = f(x³+2x+3) = 2(x³+2x+3)+3 = 2x³+4x+9

5. f(fg(x)) = f(f(x)*g(x)) = f((2x+3)(-x³)) = f(-2x⁴-3x³) = 2(-2x⁴-3x³)+3 = -4x⁴-6x³+3

Answer 6

Hint:

1/

g(g(x)) = g(-x^3) = ...

3/

f((f+g)(x) )=f(2x+3-x^3) = ...