Composition of Function (2024)

In this lesson, I will go over eight (8) worked examples to illustrate the process involved in function composition.

If we are given two functions, it is possible to create or generate a “new” function by composing one into the other. The step involved is similar when a function is being evaluatedfor a given value. For instance, evaluate the function below for [latex]x = 3[/latex].

Composition of Function (1)

It is obvious that I need to replace each [latex]x[/latex] with the given value and then simplify.

Composition of Function (2)

The key idea in function composition is that the input of the function is not a numerical value, instead, the input is also another function.

General Rule of Composition of Function

Suppose the two given functions are [latex]f[/latex] and [latex]g[/latex], the composition of [latex]f \circ g[/latex] is defined by

Composition of Function (3)

Also, the composition of [latex]g \circ f[/latex] is defined by

Composition of Function (4)

Few notesabout the symbolic “formula” above:

  • The order in function composition matters! You always compose functions from right to left. Therefore, given a function, its input is always the one to its right side. In other words, the right function goes inside the left function.
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  • Notice in [latex]f \circ g = f\left[ {g\left( x \right)} \right][/latex] , the input or “inner function” is function [latex]g[/latex] because it is to the right of function [latex]f[/latex]which is the main or “outer function”.
  • In terms of the order of composition, do you see the same pattern in[latex]g \circ f = g\left[ {f\left( x \right)} \right][/latex] ? That’s right! The function [latex]f[/latex] is the inner function of the outer function [latex]g[/latex].

Let us go over a few examples to see how function composition works. You will realize later that it is simply an exercise of algebraic substitution and simplification.

Examples of How to Compose Functions

Example 1: Perform the indicated function composition:

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The order of composition is important. Notice that in [latex]f \circ g[/latex], we want the function [latex]g\left( x \right)[/latex] to be the input of the main function [latex]{f\left( x \right)}[/latex].

It should look like this:

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I start by writing down the main or outer function [latex]f\left( x \right)[/latex], and in every instance of [latex]x[/latex], I will substitute the full value of [latex]g\left( x \right)[/latex].

Then, I’ll do whatever is needed to simplify the expressions such as squaring the binomial, applying the distributive property, and combining like terms. Other than that, there’s really nothing much to it.

Let me show you what I meant by that.

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Example 2: Perform the indicated function composition:

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I need to find the composite function [latex]g \circ f[/latex] which means function [latex]f[/latex] is the input of function [latex]g[/latex].

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Example 3: Perform the indicated function composition:

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This is an example of function composition where the input is a square root function. Let’s see how it works out.

Again, in [latex]f \circ g[/latex] we want to plug in function [latex]g[/latex] into the function [latex]f[/latex].

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Example 4: Perform the indicated function composition:

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This composition of function is quite interesting. I hope you can see that we will have a situation where a square root function goes inside another square root function.

The key to correctly composing this function is to recognize that the square root symbol can be expressed as an exponential expression with a fractional exponent equaling to [latex]{1 \over 2}[/latex].

Thus, we have [latex]\sqrt {x – 1} = {\left( {x – 1} \right)^{{1 \over 2}}}[/latex] and [latex]\sqrt x = {\left( x \right)^{{1 \over 2}}}[/latex].

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Example 5: Perform the indicated function composition:

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So far in our previous examples, we have performed function compositions using two distinct functions. However, it is also possible to compose a function with itself.

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Example 6: Perform the indicated function composition:

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Let us work out an example of a function composition that deals with rational functions. The algebra involved is a bit tedious, however, you should be okay as long as you are careful in simplifying the expressions in every step of the way.

In this example, you willapply the procedures on how to add or subtractrational expressions, and also on how to multiply rational expressions.

Here we go…

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That wasn’t too bad, right?

Example 7: Perform the indicated function composition:

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If you think that our last example of rational function composition was messy, wait till you see this next example. It can be a bit messier but still very manageable. So don’t fret! Always have that “laser” focus in everysimplification process in order to successfully work this out correctly.

The input function [latex]f[/latex] will be substituted into every [latex]x[/latex] of the main function [latex]g[/latex].

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That was easy, wasn’t it?

For more practice, I suggest that you try reversing the order of function composition. In other words, find [latex]f \circ g[/latex].

Composition of Function (21)

Do you also get?

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If that’s the case where [latex]g \circ f = f \circ g = x[/latex],then we conclude that functions [latex]g[/latex] and [latex]f[/latex] are inverses of each other. I have a separate tutorial on how to prove or verify if two functions are inverse of each other.

Example 8: Find the composite function:

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In this example, we are going to compose three functions. Observing the notation of the desired composite function [latex]f \circ g \circ h[/latex], we are going to work it out from right to left.

I first need to plug in function [latex]h[/latex] into function [latex]g[/latex] then simplify to get a new function.

The output of the previous step will be substituted further into the main function [latex]f[/latex] to obtain the final answer. Symbolically, it looks like this…

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Here we go…

I will start by finding the composition [latex]g \circ h = g\left( h \right)[/latex].

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The result of [latex]g\left( h \right) = {\Large{{x \over {{x^2} + 1}}}}[/latex]becomes the input of function [latex]f[/latex]

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You might also like these tutorials:

  • Evaluating Function
  • How to Determine if a Function is Even, Odd or Neither
  • Verifying if Two Functions are Inverses of Each Other

Tags: Advanced Algebra, Lessons

Composition of Function (2024)

FAQs

Composition of Function? ›

The composition of functions is always associative—a property inherited from the composition of relations. That is, if f, g, and h are composable, then f ∘ (g ∘ h) = (f ∘ g) ∘ h. Since the parentheses do not change the result, they are generally omitted. can be defined on the interval [−3,+3].

What is the definition of a composite function? ›

A composite function is a complex function created by two or more functions. The output of the inner function becomes the input of the outer function.

Which best describes what a composition of functions is? ›

Given two functions, we can combine them in such a way so that the outputs of one function become the inputs of the other. This action defines a composite function.

What is an example of a functional composition? ›

Composing function calls

For example, suppose we have two functions f and g, as in z = f(y) and y = g(x). Composing them means we first compute y = g(x), and then use y to compute z = f(y). Here is the example in the C language: float x, y, z; // ... y = g(x); z = f(y);

What is a real life example of composition of functions? ›

An example of a real-life composite function is where you're calculating the amount of money you'll have to spend on petrol. The first function will calculate the amount of petrol required by dividing the distance to be travelled (input variable) by the distance that the car can travel per litre(constant).

How do you find the composition of a function? ›

The composition of two functions g and f is the new function we get by performing f first, and then performing g. For example, if we let f be the function given by f(x) = x2 and let g be the function given by g(x) = x + 3, then the composition of g with f is called gf and is worked out as gf(x) = g(f(x)) .

How can you tell if a function is composite? ›

A composite function of two functions combines the given two functions in the given order. i.e., for any given two functions f(x) and g(x), there can be 4 composite functions: f(g(x)) which is substituting g(x) into f(x) g(f(x)) which is substituting f(x) into g(x)

What is the composition of a function and itself? ›

Composing a Function with Itself To compose a function with itself, we simply input a function into itself using the definition of composition of functions. In other words, to compose a function, , with itself, we compute f ( f ( x ) ) or ( f ∘ f ) ( x ) .

What is the composition of a function set? ›

In mathematics, the composition of a function is a step-wise application. For example, the function f: A→ B & g: B→ C can be composed to form a function which maps x in A to g(f(x)) in C. All sets are non-empty sets. A composite function is denoted by (g o f) (x) = g (f(x)).

What is the composition of a function and relation? ›

The composition of a function is a process in which two functions, f and g, are combined to produce a new function, h, with the formula h(x) = g(f(x)). It means that the g function is being applied to the x function. In other words, a function is applied to the output of another function.

What is an example of composition? ›

What is an example of a composition? A composition about the benefits of wind power might present the benefits and drawbacks of wind power. A composition presents a thesis (an assertion plus reasons) about the topic.

What subject is composition of functions? ›

In mathematics, a function is like a machine. It performs a set of operations on an input in order to produce an output. Therefore, a composition of functions occurs when the output, or result, of one function becomes the input of another function.

What is function composition standard? ›

The function composition f o g denotes the function with the property that (f o g) (x) = f(g x). This form of composition is known as composition in functional order. Another form of composition defines g; f to be f o g. This is known as composition in diagrammatic order.

What is a real life example of composition? ›

Composition implies a relationship where the child cannot exist independent of the parent. Example: House (parent) and Room (child). Rooms don't exist separate to a House.

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