Composition of Functions
Let A B and C be three non-empty sets. Let f →A : B and g →B : C be two mappings (or functions), then gof → A : C. This function is called the product or composite of f and g given by
gof(x) = g{f(x)} ∀ x $\in$ A
gof exists iff the range of f is a subset of domain of g. Similarly, fog exists if range of g is a subset of domain of f .
Properties of Composition of Function
(i) The composition of functions is not commutative.
i.e., fog $\neq$ gof
(ii) The composition of functions is associative.
i.e., fo(goh) = (fog)oh
(iii) The composition of any function with the identity function is the function itself.
i.e., If f : A → B, then fo$I_A$ = $I_B$of = f
Sample Problem 1
If f (x) = 5x and g(x) = x+3, then find (f∘g)(x) if x = 2.
Answer: Given, f(x) = 5x
g(x) = x+ 3
Therefore, the composition of f from g will be;
(f∘g)(x) = f(g(x)) = f(x+3) = 5(x+3)
Now putting the value of x = 2
f(g(2)) = 5(2+3) = 5(5) = 25
Sample Problem 2
If f(x) = x +1 and g(x) = -2x2, then find (g∘f)(x) for x = 1.
Answer: Given,
f(x) = x+1
g(x) = -2x2
To find: g(f(x))
g(f(x)) = g(x+1) = -2(x+1)2
Now put x =1 to get;
g(f(1)) = -2(1+1)2
= -2(2)2
= -8
Sample Problem 3
If f →R : R, g →R : R and h →R : R are such that
f(x)= x$^2$, g(x) = tan x and h(x) = log x, then the
value of (ho(gof))(x) , if x =
$\frac{\pi}{4}$
(a) 0
(b) 1
(c) –1
(d) p
Answer: (a)
(ho(gof))(x) = ho{g(x$^2$)}= ho(tan x$^2$) = log (tan x$^2$)
at x = $\frac{\pi}{4}$
(ho(gof))(x) = log $\left(tan\left(\frac{\pi}{4}\right)^2\right)$
(ho(gof))(x) = log $\left(tan \frac{\pi}{4}\right)$ = log 1 = 0
Sample Problem 4
If there are three functions, such as f(x) = x, g(x) = x2 and h(x) = 4x. Then find the composition of these functions such as [f ∘ (g ∘ h)] (x) for x = -2.
Solution: Given,
f(x) = x, g(x) = x2 and h(x) = 4x
To find: [f ∘ (g ∘ h)] (x)
[f ∘ (g ∘ h)] (x) = f ∘ (g(h(x)))
= f ∘ g(4x)
= f((4x)2)
= f(16x2)
= 16x2
If x = -2, then;
[f ∘ (g ∘ h)] (-2) = 16(-2)2 = 64
Sample Problem 5
Suppose f(x) gives miles that can be driven in x hours and g(y) gives the gallons of gas used in driving y miles. Which of these expressions is meaningful: f(g(y)) or g(f(x))?
Answer:
The function y = f(x)
is a function whose output is the number of miles driven corresponding to the number of hours driven.
The function g(y)
is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:
The expression g(y)
takes miles as the input and a number of gallons as the output. The function f(x) requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression
is meaningless.
The expression f(x)
takes hours as input and a number of miles driven as the output. The function requires a number of miles as the input. Using f(x) (miles driven) as an input value for g(y), where gallons of gas depends on miles driven, does make sense. The expression g(f(x)) makes sense, and will yield the number of gallons of gas used, driving a certain number of miles, f(x) in x hours.
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