Composite Functions

When we are given two functions, we can combine them to create a third function by composing one into the other. The procedures required to perform this operation are similar to those required to solve any function for any given value. Of that kind, functions are referred to as composite functions.

In general, a composite function is one that is written inside another function. The composition of a function has been accomplished by substituting one function for another.

For instance, f [g (x)] is the product of f (x) and g. (x). The composite function f [g (x)] can be translated as “f of g of x.” The function g (x) is referred to as an inner function, while the function f (x) is referred to as an outer function. As a result, we can read f [g (x)] as “the function g is the inner function of the function f.”

How to Solve Composite Functions?

Whenever we use BODMAS, we always start by simplifying whatever is within the brackets. Quite, in order to find f(g(x)), g(x) must first be calculated and then substituted within f. (x). Similarly, in order to find g(f(x)), f(x) must first be calculated and then substituted in g. (x). In other words, order is important when determining the composite functions. It follows that f(g(x)) may not be equal to g(f(x)). We find the composite function f(g(a)) for any two functions f(x) and g(x) by performing the following steps:

• Substitute x = an in g(x) to find g(a).
• Substitute x = g(a) in f(x) to find f(g(a)).

Composite Function From Graph

To discover the composite function of two non-algebraically defined functions shown graphically, remember that if (x, y) is a point on a function f(x), then f(x) = y.

• Determine g(a) first (i.e., the y-coordinate on the graph of g(x) that corresponds to x= a)
• Determine f(g(a)) (i.e., the y-coordinate on the graph of f(x) that corresponds to g(a))

Composite Function From Table

When given a graph of functions, we have already seen how to find the composite function. Tables are often used to display the points on a function graph. As a result, we follow the same steps as described in the previous section.

FAQs

How do you find the composition of functions?

To assess a composite function f(g(x)) at some x = a, compute g(a) first by substituting x = a in the function g(x). Consequently, by substituting x = g(a), insert g(a) into the function f(x). We can calculate g(f(a)) in the same way.

Q. How do you find composite function from a graph?

Ans: In order to discover the composite function of two non-algebraically defined functions shown graphically, remember that if (x, y) is a point on a function f(x), then f(x) = y.

• Determine g(a) first (i.e., the y-coordinate the on the graph of g(x) that corresponds to x= a).
• Determine f(g(a)) (i.e., the y-coordinate on the graph of f(x) that corresponds to g(a)).

Q. f(x) = 3x². find (f∘f)(x).

Ans: Given that,

f(x) = 3x²

(f∘f)(x) = f(f(x))

= f (3x²)

= 3(3x)²

= 3.9x²

= 27x²

For more visit Fractional Exponents – Explanation, Different Functions, and Solved Examples