Composite Functions

Composite functions are operations that take two or more functions as one function such as h(x) = g(x)This is mainly to do with taking numbers from one set to another set.

Composite Functions Composite Functions

Create learning materials about Composite Functions with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Contents
Table of contents

    So, for example, if a function took a number from set A to B and another function took a number from set B to C, the composite would take a number from set A directly to C.

    Here is a diagram showing how the functions f(x) and g(x) can transform an input to an output.

    Composite functions, An image which illustrates composite functions, StudySmarterAn image which illustrates composite functions

    Values in sets can be mapped to other sets by composite functions.

    Properties of composite functions

    There are important properties of composite functions that we should remember:

    Property nameDefinition
    AssociativeIf functions are composable, they are always associative. This means that no matter where the brackets are located in a function, it makes no difference to the overall function result. Therefore if f, g, h are composable then \(f(gh(x)) = (fg)h(x)\).
    CommutativeIf functions are composable it means they are not necessarily commutative. Commutativity is when swapping the order of the function's composition, does not affect it, e.g. (ab=ba).
    One-to-oneA one-to-one composite function is where there is a single output for every input. There is also a many-to-one function where many inputs can render the same output. In the definition of a function, no composite function can be one-to-many.
    InverseAn inverse must exist for a composite function, therefore, there cannot be an output for which there does not exist an input.

    How do we find a composite function?

    Essentially, we are carrying out a function of a function. Let's say we're trying to find \(h(x) = fg(x)\).

    We would first take g(x) (the output of x) and then use it as an input in f(x) hence getting fg(x). Let's look at a worked example.

    Examples of composite functions

    \(f(x) = 3x + 2\) and \(g(x) = 5x -1\). If \(h(x) = fg(x)\), find the value of h(2).

    Steps

    Example

    Step 1: Rewrite h (x).

    \(h(2) = fg(2)\)

    Step 2: Find the output of the inside function first.

    \(g(2) = 5(2) - 1 = 9\)

    Step 3: Substitute this newly found output as the input in the outside function.

    \(f(9) = 3(9) + 2 = 29\)

    FINAL ANSWER\(h(2) = 29\)
    \(f(x) = 3x + 2\) and \(g(x) = 5x -1\). If \(h(x) = fg(x)\), find h(x).

    Steps

    Example

    Step 1: Rewrite h (x).

    \(h(x) = fg(x)\)

    Step 2: Find the output of the inside function first.

    \(g(x) = 5x -1\)

    Step 3: Substitute this newly found output as the input in the outside function.

    \(f(5x-1) = 3(5x-1) +2 = 15x - 3 + 2 = 15x -1\)

    FINAL ANSWER\(h(x) = 15x -1\)

    \(f(x) = 3x +2\). Find \(f^2(x)\).

    Steps

    Example

    Step 1: rewrite \(f^2(x)\).

    \(f^2(x) = ff(x)\)

    Step 2: Find the output of the inside function first.

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

    Step 3: Substitute this newly found output as the input in the outside function.

    \(f(3x +2) = 3(3x+2) + 2 = 9x + 6 + 2 = 9x+8\)

    FINAL ANSWER\(f^2(x) = 9x+8\)

    What are some harder composite functions?

    Sometimes quadratic, trigonometric and reciprocal functions can come into play, however, the logic is exactly the same as with the easier linear examples we have seen above. Let's look at some more worked examples.

    \(f(x) = \cos(x), \space g(x) = 3x -2. \space h(x) = gf(x)\). Find the value of h(90).

    steps

    Example

    Step 1: Rewrite h (x).

    \(h(90) = gf(90)\)

    Step 2: Find the output of the inside function first.

    \(f(90) = \cos(90) = 0\)

    Step 3: Substitute this newly found output as the input in the outside function.

    \(g(0) = 3(0) - 2 = 0-2 = -2\)

    FINAL ANSWER \(g(0) = -2\)

    \(f(x) = \tan^{-1}(x)\), with \(0 \leq tan^{-1}(x) \leq 2 \pi\ ), \(g(x) = x^2 + 6x -8\). \(h(x) = gf(x)\). Find the value of h(1).

    steps

    Example

    Step 1: Rewrite h (x).

    \(h(1) = gf(1)\)

    Step 2: Find the output of the inside function first.

    \(\tan^{-1}(1) = \frac{\pi}{4}\)

    Step 3: Substitute this newly found output as the input in the outside function.

    \(\big(\frac{\pi}{4} \big)^2 + 6 \big( \frac{\pi}{4} \big) - 8 = -2.67076074455\)

    FINAL ANSWER\(h(1) = -2.67076074455\)

    \(f(x) = \tan^{-1}(x), \space g(x) = 3-x^2. \space h(x) = gf^{-1}(x)\). Find \(h(x)\)

    steps

    Example

    Step 1: Rewrite h (x).

    \(h(x) = gf^{-1}(x)\)

    Step 2: Find the output of the inside function first.

    \(f^{-1}(x) = \tan(x)\)

    Step 3: Substitute this newly found output as the input in the outside function.

    \(g(\tan(x)) = 2 -\tan^2(x)\)

    FINAL ANSWER

    \(h(x) = 3 - \tan^2(x)\)

    Composite Functions - Key takeaways

    • Composite functions are when a combination of functions are undertaken to give a common output.
    • Always perform the inside function first before inputting this value into the outside function.
    Frequently Asked Questions about Composite Functions

    What is a composite function?

    A composite function is a composition of two or more functions.

    How do you find the domain of a composite function?

    By looking at the domain of the first function applied.

    How do we compose two functions?

    By applying a singular input and finding the output of the functions put together.

    How do we find the value of a composite function?

    By getting an output from inputing a value into the original function, then applying all the other functions to this output value.

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 5 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner