COMPOSITION OF FUNCTIONS FROM THE GRAPH

Example 1 :

Using the graphs below, evaluate f (g(1))

Solution :

f (g(1))

The value of g(1) from the graph of g(x) is 3.

So, f (g(1)) = f(3)

Value of f(3) from the graph f(x) is 6.

Then, the value of f (g(1)) is 6.

Example 2 :

Use the graphs to evaluate the expressions below.

(i)  f (g(3))    (ii)  f (g (1))    (iii)  g( f (1))      (iv)  g ( f (0))

(v) f( f (5))     (vi)  f ( f (4))     (vii)  g(g(2))    (viii) g ( g (0))

Solution :

(i)  f (g(3))

From the graph of g(x), first we find g(3).

g(3) = 4

f (g(3)) = f(4)

From the graph f(x), we find f(4).

The value of f(4) is 0. So, f(g(3)) = 0.

(ii)  f (g (1))

Value of g(1) = 3, then f (g (1)) = f(3)

From the graph of f(x), the value of f(3) is 2.

(iii)  g( f (1))

Value of f(1) = 3, then g (f (1)) = g(3)

From the graph of g(x), the value of g(3) is 4.

(iv)  g ( f (0))

Value of f(0) = 4, then g (f (0)) = g(4)

From the graph of g(x), the value of g(4) is 5.

(v) f( f (5))

Value of f(5) = 1, then f (f (5)) = f(1)

From the graph of f(x), the value of f(1) is 3.

(vi)  f ( f (4))

Value of f(4) = 0, then f (f (4)) = f(0)

From the graph of g(x), the value of f(0) is 4.

(vii)  g(g(2))

Value of g(2) = 0, then g (g (2)) = g(0)

From the graph of g(x), the value of g(0) is 2.

(viii) g ( g (0))

Value of g(0) = 2, then g (g (0)) = g(2)

From the graph of g(x), the value of g(2) is 0.

Example 3 :

Use the graphs to evaluate the composition of functions.

(i)  (g ∘ f)(1)

(ii)  (g ∘ f)(5)

(iii)  (f ∘ g)(0)

(iv)  (f ∘ g)(2) 

Solution :

(i)  (g ∘ f)(1) = g[f(1)]

The value of f(1) is -1.

g[f(1)] = g(-1)

The value of g(-1) is 3.

(g ∘ f)(1) = 3

(ii)  (g ∘ f)(5) = g[f(5)]

The value of f(5) is 1.

g[f(5)] = g[1]

The value of g(1) is 4.

(g ∘ f)(5) = 4

(iii)  (f ∘ g)(0) = f[g(0)]

The value of g(0) is 5.

f[g(0)] = f(5)

The value of f(5) is 1.

(iv)  (f ∘ g)(2) = f[g(2)]

The value of g(2) is 2.

f[g(2)] = f(2)

The value of f(2) is -2.

(f ∘ g)(2) = -2

Example 4 :

Use the graph to find each value.

Find 

a) f( g (2) )

b)  (g ∘ f) (3)

c) g ( f (2) )

Solution :

a) f( g (2) ) = f[g(2)]

When input is 2, for the function g(x) the output is 3

= f [3]

When input is 3 for the function f(x), the output is -1

= -1

So, the value of  f( g (2) ) is -1.

b)  (g ∘ f) (3) = g [ f(3) ]

When input is 3, for the function f(x) the output is -1

= g [-1]

When the input is -1, the output for the function g(x) is 0.

= 0

c) g ( f (2) ) = g [ f(2) ]

When input is 2, the output for the function f(x) is -2

= g[-2]

When input is -2, the output for the function g(x) is 0.

= 0

Example 5 :

Find 

a) f( g(2) )

b)  (f ∘ g) (-1)

c) g ( f(1) )

Solution :

a)

f( g(2) ) = f[g(2)]

= f[-2]

= 2

b) 

(f ∘ g) (-1) = f[g(-1)]

= f(-3)

= 1

c)

g ( f(1) ) = g[f(1)]

= g[3]

= -3

The first of the two graphs shows two functions 𝑓 𝑎𝑛𝑑 𝑔. The second shows two functions ℎ 𝑎𝑛𝑑 𝑘. Use the graphs to compute the following:

a)

(𝑔 ◦ 𝑓) (−4) = g[f(-4)]

= g[0]

= -3

b)

(𝑓 ◦ 𝑔) (3) = f[g(3)]

= f [0]

= 2

c)

(𝑓 ◦ 𝑓) (−2) = f[f(-2)]

= f[3]

= 1

d)

(𝑔 ◦ 𝑔) (3) = g[g(-3)]

= g[-5]

= undefined

e)

(𝑔 ◦ 𝑓) (−5) = g[f(-5)]

= g[-3]

= -5

f)

(𝑔  ◦ 𝑓) (−3) = g[f(-3)]

= g[1.5]

= -1.5

g)

(ℎ ◦ 𝑘) (0) = h[k(0)]

= h[4]

= 4