Composite & Inverse Functions (DP IB Maths: AA SL): Revision Note

Author

Dan

Last updated

17 July 2023

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Composite Functions

What is a composite function?

A composite function is where a function is applied to another function
A composite function can be denoted
- $(f \circ g) (x)$
- $f g (x)$
- $f (g (x))$
The order matters
- $(f \circ g) (x)$ means:
  - First apply g to x to get $g (x)$
  - Then apply f to the previous output to get $f (g (x))$
  - Always start with the function closest to the variable
- $(f \circ g) (x)$ is not usually equal to $(g \circ f) (x)$

How do I find the domain and range of a composite function?

The domain of $f \circ g$ is the set of values of $x$ ...
- which are a subset of the domain of g
- which maps g to a value that is in the domain of f
The range of $f \circ g$ is the set of values of $x$ ...
- which are a subset of the range of f
- found by applying f to the range of g
To find the domain and range of $f \circ g$
- First find the range of g
- Restrict these values to the values that are within the domain of f
  - The domain is the set of values that produce the restricted range of g
  - The range is the set of values that are produced using the restricted range of g as the domain for f
For example: let $f (x) = 2 x + 1, - 5 \leq x \leq 5$ and $g (x) = \sqrt{x}, 1 \leq x \leq 49$
- The range of g is $1 \leq g (x) \leq 7$
  - Restricting this to fit the domain of f results in $1 \leq g (x) \leq 5$
- The domain of $f \circ g$ is therefore $1 \leq x \leq 25$
  - These are the values of x which map to $1 \leq g (x) \leq 5$
- The range of $f \circ g$ is therefore $3 \leq (f \circ g) (x) \leq 11$
  - These are the values which f maps $1 \leq g (x) \leq 5$ to

Examiner Tip

Make sure you know what your GDC is capable of with regard to functions
- You may be able to store individual functions and find composite functions and their values for particular inputs
- You may be able to graph composite functions directly and so deduce their domain and range from the graph
$f f (x)$ is not the same as ${[f (x)]}^{2}$

Worked example

Given $f (x) = \sqrt{x + 4}$ and $g (x) = 3 + 2 x$ :

Write down the value of

(g \circ f) (12)

2-3-2-ib-aa-sl-composite-functions-a-we-solution

Write down an expression for

(f \circ g) (x)

2-3-2-ib-aa-sl-composite-functions-b-we-solution

Write down an expression for

(g \circ g) (x)

2-3-2-ib-aa-sl-composite-functions-c-we-solution

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Inverse Functions

What is an inverse function?

Only one-to-one functions have inverses
A function has an inverse if its graph passes the horizontal line test
- Any horizontal line will intersect with the graph at most once
The identity function $id$ maps each value to itself
- $id (x) = x$
If $f \circ g$ and $g \circ f$ have the same effect as the identity function then $f$ and $g$ are inverses
Given a function $f (x)$ we denote the inverse function as $f^{- 1} (x)$
An inverse function reverses the effect of a function
- $f (2) = 5$ means $f^{- 1} (5) = 2$
Inverse functions are used to solve equations
- The solution of $f (x) = 5$ is $x = f^{- 1} (5)$
A composite function made of $f$ and $f^{- 1}$ has the same effect as the identity function
- $(f \circ f^{- 1}) (x) = (f^{- 1} \circ f) (x) = x$

What are the connections between a function and its inverse function?

The domain of a function becomes the range of its inverse
The range of a function becomes the domain of its inverse
The graph of $y = f^{- 1} (x)$ is a reflection of the graph $y = f (x)$ in the line $y = x$
- Therefore solutions to $f (x) = x$ or $f^{- 1} (x) = x$ will also be solutions to $f (x) = f^{- 1} (x)$
  - There could be other solutions to $f (x) = f^{- 1} (x)$ that don't lie on the line $y = x$

Inverse Functions Notes Diagram 2

How do I find the inverse of a function?

STEP 1: Swap the x and y in $y = f (x)$
- If $y = f^{- 1} (x)$ then $x = f (y)$
STEP 2: Rearrange $x = f (y)$ to make $y$ the subject
Note this can be done in any order
- Rearrange $y = f (x)$ to make $x$ the subject
- Swap $x$ and $y$

Examiner Tip

Remember that an inverse function is a reflection of the original function in the line $y = x$
- Use your GDC to plot the function and its inverse on the same graph to visually check this
$f^{- 1} (x)$ is not the same as $\frac{1}{f (x)}$

Worked example

For the function $f (x) = \frac{2 x}{x - 1}, x > 1$ :

Find the inverse of

f (x)

2-3-2-ib-aa-sl-inverse-functions-a-we-solution

Find the domain of

f^{- 1} (x)

2-3-2-ib-aa-sl-inverse-functions-b-we-solution

Find the value of

k

such that

f (k) = 6

2-3-2-ib-aa-sl-inverse-functions-c-we-solution

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Previous:2.3.1 Language of FunctionsNext:2.3.3 Graphing Functions

Composite & Inverse Functions (DP IB Maths: AA SL): Revision Note

What is a composite function?

How do I find the domain and range of a composite function?

What is an inverse function?

What are the connections between a function and its inverse function?

How do I find the inverse of a function?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Proof & Reasoning

1.4.1 Proof

1.5 Binomial Theorem

1.5.1 Binomial Theorem

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Graphing Functions

2.4 Further Functions & Graphs

2.4.1 Reciprocal & Rational Functions

2.4.2 Exponential & Logarithmic Functions

2.4.3 Solving Equations

2.4.4 Modelling with Functions

2.5 Transformations of Graphs

2.5.1 Translations of Graphs

2.5.2 Reflections of Graphs

2.5.3 Stretches of Graphs

2.5.4 Composite Transformations of Graphs

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Radian Measure

3.1.3 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Further Trigonometry

3.4.1 The Unit Circle

3.4.2 Exact Values

3.5 Trigonometric Functions & Graphs

3.5.1 Graphs of Trigonometric Functions

3.5.2 Transformations of Trigonometric Functions

3.5.3 Modelling with Trigonometric Functions

3.6 Trigonometric Equations & Identities

3.6.1 Simple Identities

3.6.2 Double Angle Formulae

3.6.3 Relationship Between Trigonometric Ratios

3.6.4 Linear Trigonometric Equations

3.6.5 Quadratic Trigonometric Equations

4. Statistics & Probability

4.1 Statistics Toolkit

4.1.1 Sampling & Data Collection

4.1.2 Statistical Measures

4.1.3 Frequency Tables