Sign In | Starter Of The Day | Tablesmaster | Fun Maths | Maths Map | Topics | More

International Baccalaureate Mathematics

Functions

Syllabus Content

Composite functions. Identity function. Finding the inverse function f-1(x)

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

Here are some exam-style questions on this statement:

See all these questions

Here is an Advanced Starter on this statement:

Click on a topic below for suggested lesson Starters, resources and activities from Transum.


Furthermore

Official Guidance, clarification and syllabus links:

\((f \circ g)(x)=f(g(x)) \)

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

The existence of an inverse for one-to-one functions.

Link to: concept of inverse function as a reflection in the line \(y=x\) (SL 2.2).

Composite functions involve combining two or more functions to create a new function. The identity function, denoted as \( f(x) = x \), is a simple function where the output is the same as the input. Finding the inverse function, \( f^{-1}(x) \), involves swapping the x and y values in the original function and solving for y, which represents the new function \( f^{-1}(x) \).

The key formula for a composite function where two functions \( f \) and \( g \) are combined is given by: $$ (f \circ g)(x) = f(g(x)) $$ To find the inverse function \( f^{-1}(x) \), the equation \( y = f(x) \) is rearranged to make \( x \) the subject: $$ x = f^{-1}(y) $$

Example:
Let \( f(x) = 2x + 3 \) and \( g(x) = x - 1 \). The composite function \( (f \circ g)(x) \) and the inverse function \( f^{-1}(x) \) are found as follows:
Composite function: $$ (f \circ g)(x) = f(g(x)) = f(x - 1) = 2(x - 1) + 3 = 2x + 1 $$ Inverse function of \( f \): Given \( y = 2x + 3 \), swap \( x \) and \( y \) and solve for \( y \): $$ x = 2y + 3 \Rightarrow y = \frac{x - 3}{2} $$ So, \( f^{-1}(x) = \dfrac{x - 3}{2} \).

Transum,

Saturday, August 17, 2019

"There is an Advanced Lesson Starter called Permutable Functions that is open ended but allows students to consolidate their understanding of composite functions."

How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers anywhere in the world. Click here to enter your comments.


Apple

©1997-2024 WWW.TRANSUM.ORG