In (f ∘ g) (x),
(f ∘ g) (x) = f[g(x)]
Here g(x) is the input for the function f(x).
Wherever we see x in the function f(x), we replace it by the function g(x).
Example :
(f ∘ g) (x) = 1/(x - 3)
Solution :
In the given composition function, in the place of x, we see x - 3.
Since x - 3 is the replacement, g(x) = x - 3
Then,
f(x) = 1/x
Finding the composition of function (f ∘ g) (x), the result matches with H(x). So, the required functions are
f(x) = 1/x and g(x) = x - 3
Note :
We can find more than one possible functions for the given composition of function.
Find functions f and g so that f∘g = H
Problem 1 :
H(x) = (2x + 3)^{4}
Solution :
Given that,
(f∘g)(x) = H (x)
= f [ g(x) ]
From this, we can understand that where we see x in the function f(x), it is being replaced by the function g(x).
Let g(x) = 2x + 3, then f(x) must be x^{4}
Verifying our assumption :
= f(2x + 3)
= (2x + 3)^{4}
So, the required functions f and g are
f(x) = x^{4 }and g(x) = 2x + 3
Problem 2 :
H(x) = (1+x^{2})^{3}
Solution :
Given that,
(f∘g)(x) = H (x)
= f [ g(x) ]
One possible answer :
If g(x) = 1 + x^{2}, then f(x) = x^{3}
Other possible answers are also there :
If g(x) = x^{2}, then f(x) = (1 + x)^{3}
Problem 3 :
H(x) = √(x^{2} + 1)
Solution :
Given that,
(f∘g)(x) = H (x)
= f [ g(x) ]
One possible answer :
If g(x) = √(x + 1), then f(x) = x^{2}
Other possible answer :
If g(x) = √x, then f(x) = x^{2 }+ 1
Problem 4 :
H(x) = |2x + 1|
Solution :
Given that,
(f∘g)(x) = H (x)
= f [ g(x) ]
One possible answer :
If g(x) = |x|, then f(x) = 2x + 1
One possible answer :
If g(x) = |x - 1|, then f(x) = 2(x + 1)
Problem 5 :
H(x) = |2x^{2} + 3|
Solution :
Given that,
(f∘g)(x) = H (x)
= f [ g(x) ]
One possible answer :
If g(x) = |2x+3|, then f(x) = x^{2}
Problem 6 :
Most functions are compositions of basic functions. Work backwards to determine the basic functions that created the composition
Solution :
Let H(x) = f(g(x))
a) f(g(x)) = (x + 4)^{2} + 5
H(x) = (x + 4)^{2} + 5
If f(x) = x^{2} + 5 and g(x) = x + 4
b) f(g(x)) = √(x - 4)
H(x) = √(x - 4)
If g(x) = x - 4, then f(x) = √x
c) f(g(x)) = (4x - 1)^{2}
H(x) = (4x - 1)^{2}
If g(x) = 4x - 1, then f(x) = x^{2}
d) f(g(x)) = |x + 2|
H(x) = |x + 2|
If g(x) = (x + 2), then f(x) = |x|
e) f(g(x)) = √x - 4
H(x) = √x - 4
If g(x) = √x, then f(x) = x - 4
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM