UPSKILL MATH PLUS
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Learn moreGiven two functions f(x) and g(x), then the composition f \circ g is computed as follows:
Working rule:
Step 1: Rewrite the given composition as (f \circ g)(x) = f\left(g(x)\right).
Step 2: Using the individual functions as a reference, replace the variable x in the outside function with the inside function.
Step 3: Simplify the resulting function.
Example:
If f(x) = x -6 and g(x) = x^2, then find f \circ g.
Solution:
(f \circ g)(x) = f\left(g(x)\right)
= f\left(x^2\right)
= x^2 -6
- A function can also be composed of itself. If f is a function, then its composition with itself is given by (f \circ f)(x) = f\left(f(x)\right).
- The composition of functions is not commutative. That is, f \circ g \neq g \circ f.
Important!
Example:
If f(x) = x - 6 and g(x) = x^2, then check whether the composition of the two functions are commutative.
Solution:
First, let us find (f \circ g).
(f \circ g)(x) = f\left(g(x)\right)
= f\left(x^2\right)
= x^2 -6
Now, let us find (g \circ f).
(g \circ f)(x) = g\left(f(x)\right)
= g\left(x - 6\right)
= \left(x -6\right)^2
= x^2 - 12x + 36
It is observed that f \circ g \neq g \circ f.
Therefore, the composition of the two functions f and g is not commutative.
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