The composition of functions is the process of combining two or more functions into a single function. The symbol of the composition of functions is ∘.
Let us see an example to see how to find composition of two functions.
Let the two functions be
f(x) = 2x - 7 and g(x) = 3x + 5
find (f∘g)(x).
Finding (f∘g)(x) :
(f∘g)(x) = f[g(x)]
= f[3x + 5]
We consider this function as f(x). Instead of x we have 3x + 5. So, in the function f(x), replace x by 3x + 5.
f(3x + 5) = 2(3x + 5) - 7
= 6x + 10 - 7
= 6x + 3
Example 1 :
Given f(t) = t2 - t and h(x) = 3x + 2 , evaluate (f ∘ h)(1).
Solution :
(f ∘ h)(1) = f[h(1)] ----(1)
Let us evaluate h(1).
h(x) = 3x + 2
h(1) = 3(1) + 2
h(1) = 5
By applying the value of h(1) = 5 in (1) , we get
(f ∘ h)(1) = f(5)
Finding the value of f(5), we get
f(5) = 52 - 5
f(5) = 25 - 5
f(5) = 20
Example 2 :
Given each pair of functions
f(x) = 4x + 8, g(x) = 7 - x2
calculate
(i) f (g (0)) and (ii) g ( f (0))
Solution :
f(x) = 4x + 8, g(x) = 7 - x2
Evaluating f (g (0)) :
(i) f (g (0)) -----(1)
g(0) = 7 - 02
g(0) = 7
Applying g(0) = 7 in (1), we get
f(g(0)) = f(7)
f(7) = 4(7) + 8
f(7) = 28 + 8
f(7) = 36
(ii) Evaluating g (f (0)) :
g ( f (0)) ----(2)
f(0) = 4(0) + 8
f(0) = 8
Applying f(0) = 8 in (2), we get
g(f(0)) = g(8)
g(8) = 7 - 82
= 7 - 64
= -57
Example 3 :
Given each pair of functions
f(x) = 5x + 7, g(x) = 4 - 2x2
calculate
(i) f (g (0)) and (ii) g (f (0))
Solution :
Evaluating f (g (0)) :
(i) f (g (0)) -----(1)
g(0) = 4 - 2(0)2
g(0) = 4
Applying g(0) = 4 in (1), we get
f(g(0)) = f(4)
f(4) = 5(4) + 7
f(4) = 20 + 7
f(4) = 27
(ii) Evaluating g (f (0)) :
g ( f (0)) ----(2)
f(0) = 5(0) + 7
f(0) = 7
Applying f(0) = 7 in (2), we get
g(f(0)) = g(7)
g(7) = 4 - 2(72)
= 4 - 2(49)
= 4 - 98
= -94
Example 4 :
Given that f(x) = 3x - 4 and g(x) = 7 - x2 calculate.
(i) (f ∘ g)(2)
(ii) (g ∘ f)(2)
(iii) (f ∘ g)(-2)
(iv) (g ∘ f)(-2)
Solution :
(i) (f ∘ g)(2) = f[g(2)]
g(2) = 7 - 22 = 7 - 4 = 3 |
f[g(2)] = f(3) = 3(3) - 4 = 9 - 4 (f ∘ g)(2) = 5 |
(ii) (g ∘ f)(2) = g[f(2)]
= g[3(2) - 4]
= g[6 - 4]
= g[2]
Now in the function g(x), instead of x we will apply 2.
= 7 - 22
= 7 - 4
(g ∘ f)(2) = 3
(iii) (f ∘ g)(-2) = f[g(-2)]
g(x) = 7 - x2
= f [7 - (-2)2]
= f [7 - 4]
= f [3]
Now in the function f(x), we will replace x by 3.
f(x) = 3x - 4
= 3(3) - 4
= 9 - 4
= 5
(iv) (g ∘ g)(-2) = g[g(-2)]
g[g(2)] = g[7 - (-2)2]
= g[7 - 4]
= g[3]
Now in the function g(x), instead of x we will apply 3.
g(3) = 7 - 32
= 7 - 9
= -2
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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