EVALUATING COMPOSITION OF FUNCTIONS

Use the function f and g given below to evaluate the following expressions:

f(x) = 3 - 2x and g(x) = x² - 5x + 4

Problem 1 :

(a) g(0)         (b) f(g(0))          (c) f(0)            (d) g(f(0))

Solution:

(a) g(0) :

g(0) = (0² - 5(0) + 4)

= 4

(b) f(g(0)) :

f(g(0)) = f(4)

= 3 - 2(4)

= 3 - 8

= -5

(c) f(0) :

f(0) = 3 - 2(0)

= 3 - 0

= 3

(d) g(f(0)) :

g(f(0)) = g(3)

= 3² - 5(3) + 4

= 9 - 15 + 4

= -2

Problem 2 :

(a) g(-1)      (b) f(g(-1))         (c) f(-1)          (d) g(f(-1))

Solution:

(a) g(-1) :

 g(-1) = (-1)² - 5(-1) + 4

= 1 + 5 + 4

= 10

(b) f(g(-1)) : 

 f(g(-1)) = f(10)

= 3 - 2(10)

= 3 - 20

= -17

(c) f(-1) :

f(-1) = 3 - 2(-1)

= 3 + 2

= 5

(d) g(f(-1)) :

 g(f(-1)) = g(5)

= 5² - 5(5) + 4

= 25 - 25 + 4

= 4

Problem 3 :

For f(x) = 3 - 2x and g(x) = x² - 5x + 4

(a) (f ∘ g)(-2)         (b) (g ∘ f)(-2)

Solution:

(a) (f ∘ g)(-2) :

 (f ∘ g)(-2) = f ∘ g(-2)

= f[(-2)² - 5(-2) + 4]

= f(4 + 10 + 4)

= f(18)

= 3 - 2(18)

= 3 - 36

= -33

(b) (g ∘ f)(-2) :

 (g ∘ f)(-2) = g ∘ f(-2)

= g[3 - 2(-2)]

= g(3 + 4)

= g(7)

= 7² - 5(7) + 4

= 49 - 35 + 4

= 18

Problem 4 :

For f(x) = 3 - 2x and g(x) = x² - 5x + 4

(a) (f ∘ g)(4)            (b) (g ∘ f)(4)

Solution:

(a) (f ∘ g)(4) :

 (f ∘ g)(4) = f ∘ g(4)

= f[4² - 5(4) + 4]

= f(16 - 20 + 4)

= f(0)

= 3 - 2(0)

= 3

(b) (g ∘ f)(4) :

 (g ∘ f)(4) = g ∘ f(4)

= g[3 - 2(4)]

= g(3 - 8)

= g(-5)

= (-5)² - 5(-5) + 4

= 25 + 25 + 4

= 54

Problem 5 :

For f(x) = 3 - 2x and g(x) = x² - 5x + 4

(a) (f ∘ f)(6)      (b) (g ∘ g)(6)

Solution:

(a) (f ∘ f)(6) :

(f ∘ f)(6) = f ∘ f(6)

= f[3 - 2(6)]

= f(3 - 12)

= f(-9)

= 3 - 2(-9)

= 3 + 18

= 21

(b) (g ∘ g)(6) :

(g ∘ g)(6) = g ∘ g(6)

= g[(6)² - 5(6) + 4]

= g(36 - 30 + 4)

= g(10)

= (10)² - 5(10) + 4

= 100 - 50 + 4

= 54

Problem 6 :

For f(x) = 3 - 2x and g(x) = x² - 5x + 4

(a) (f ∘ f)(-4)           (b) (g ∘ g)(-4)

Solution:

(a) (f ∘ f)(-4) :

(f ∘ f)(-4) = f ∘ f(-4)

= f[3 - 2(-4)]

= f(3 + 8)

 = f(11)

= 3 - 2(11)

= 3 - 22

= -19

(b) (g ∘ g)(-4) :

(g ∘ g)(-4) = g ∘ g(-4)

= g[(-4)² - 5(-4) + 4]

= g(16 + 20 + 4)

= g(40)

= (40)² - 5(40) + 4

= 1600 - 200 + 4

= 1404

Problem 7 :

(a) (f ∘ g)(x)             (b) (g ∘ f)(x)

Solution:

(a) (f ∘ g)(x) :

(f ∘ g)(x) = f ∘ g(x)

= f[x² - 5x + 4]

= 3 - 2(x² - 5x + 4)

= 3 - 2x² + 10x - 8

= -2x² + 10x - 5

(b) (g ∘ f)(x) :

(g ∘ f)(x) = g ∘ f(x)

= g[3 - 2x]

= (3 - 2x)² - 5(3 - 2x) + 4

= 4x² + 12x + 9 - 15 + 10x + 4

= 4x² + 22x - 2

Problem 8 :

f(x) = 3 - 2x and g(x) = x² - 5x + 4

(a) (f ∘ f)(x)          (b) (g ∘ g)(x)

Solution:

(a) (f ∘ f)(x) :

(f ∘ f)(x) = f ∘ f(x)

= f[3 - 2x]

= 3 - 2(3 - 2x)

= 3 - 6 + 4x

= 4x - 3

(b) (g ∘ g)(x) :

 (g ∘ g)(x) = g ∘ g(x)

= g[x² - 5x + 4]

= (x² - 5x + 4)² - 5(x² - 5x + 4) + 4

= x⁴ - 10x³ + 33x² - 40x + 16 - 5x² + 25x - 20 + 4

= x⁴ - 10x³ + 28x² - 15x