How to Solve Composite Functions?
Composite Function From Graph
How Does It Work?
Example of Composite Functions
Why Do We Use Composite Functions?
Notation for Composite Functions
Practice Question
Conclusion
FAQs

Composite Functions

Q: How do you read (f∘g)(x)?

( f ∘ g ) ( x ) is read as "f of g of x". This means you first apply g ( x ) g(x) g ( x ) and then use its result as the input for f ( x ) f(x) f ( x ) .

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.

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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.”

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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:

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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))

How Does It Work?

Start with g(x): First, use the input value (x) in the function g(x). This gives you an output value.
Use the Output in f(x): Take the output of g(x) and use it as the input for f(x). The final result is the output of the composite function f(g(x)).

Example of Composite Functions

Suppose:

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f(x) = x + 3
g(x) = 2x

To find f(g(x)):

Start with g(x): g(x) = 2x. For example, if x = 4, then g(4) = 2(4) = 8.
Use the result in f(x): f(g(x)) = f(8) = 8 + 3 = 11.

So, f(g(4)) = 11.

Why Do We Use Composite Functions?

Composite functions are useful because they let us combine processes. For example, in real life:

A company might calculate profits by first finding revenue using one function and then subtracting costs using another.
In physics, you might calculate speed using one function and then use that speed in another function to find energy.

Notation for Composite Functions

You’ll often see composite functions written as:

f(g(x))
Or sometimes (f ○ g)(x), which means the same thing.

Practice Question

If f(x) = x² and g(x) = x - 1, find f(g(x)).

Start with g(x) = x - 1.
Use this result in f(x) = x²: f(g(x)) = (x - 1)² = x² - 2x + 1.

Conclusion

Composite functions help us solve problems step by step by connecting two functions. Practice is the key to understanding them well, so try combining different functions and solving them step by step!

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.

What is a composite function?

A composite function is a function that is formed by combining two functions, say $f(x)$ $f$ $($ $x$ $)$ and $g(x)$ $g$ $($ $x$ $)$ , such that the output of one function becomes the input of the other. It is denoted as $(f \circ g)(x)$ $($ $f$ $\circ$ $g$ $)$ $($ $x$ $)$ or $f(g(x))$ $f$ $($ $g$ $($ $x$ $))$ .

How do you read (f∘g)(x)?

$($ $f$ $\circ$ $g$ $)$ $($ $x$ $)$ is read as "f of g of x". This means you first apply $g(x)$ $g$ $($ $x$ $)$ and then use its result as the input for $f(x)$ $f$ $($ $x$ $)$ .

Can composite functions be inverse functions?

Yes, if $f(g(x)) = x$ $f$ $($ $g$ $($ $x$ $))$ $=$ $x$ and $g(f(x)) = x$ $g$ $($ $f$ $($ $x$ $))$ $=$ $x$ , then $f(x)$ $f$ $($ $x$ $)$ and $g(x)$ $g$ $($ $x$ $)$ are inverses of each other. For example:

$f(x) = x + 2$ $f$ $($ $x$ $)$ $=$ $x$ $+$ $2$
$g(x) = x - 2$ $g$ $($ $x$ $)$ $=$ $x$ $-$ $2$
$f(g(x)) = (x - 2) + 2 = x$ $f$ $($ $g$ $($ $x$ $))$ $=$ $($ $x$ $-$ $2$ $)$ $+$ $2$ $=$ $x$
$g(f(x)) = (x + 2) - 2 = x$ $g$ $($ $f$ $($ $x$ $))$ $=$ $($ $x$ $+$ $2$ $)$ $-$ $2$ $=$ $x$

How are composite functions used in real life?

Composite functions are used to model situations where one process depends on the outcome of another. Examples include:

Physics: Calculating motion where velocity depends on time, and distance depends on velocity.
Economics: Profit depends on sales, and sales depend on price.
Biology: Enzyme activity depends on substrate concentration, which depends on temperature.

How to Solve Composite Functions?
Composite Function From Graph
How Does It Work?
Example of Composite Functions
Why Do We Use Composite Functions?
Notation for Composite Functions
Practice Question
Conclusion
FAQs

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