🎯⭐ INTERACTIVE LESSON

Function Composition & Inverses

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Function Composition & Inverses - Complete Interactive Lesson

Part 1: Function Composition

🔗 Function Composition

Part 1 of 7

What Is Composition?

The composition of $f$ and $g$ , written $(f \circ g)(x)$ , means " $f$ of $g$ of $x$ ":

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

Think of it as a pipeline: input $x$ flows into $g$ first, then the output flows into $f$ .

Example

If $f(x) = x^2$ and $g(x) = x+3$ :

$(f \circ g)(x) = f(g(x)) = f(x+3) = (x+3)^2$

$(g \circ f)(x) = g(f(x)) = g(x^2) = x^2+3$

⚠️ Order matters! $f \circ g \neq g \circ f$ in general.

📝 More Examples

Example 1

$f(x)=2x+1, g(x)=x^2-4$

🔍 Domain of a Composition

The domain of $f \circ g$ requires:

$x$ must be in the domain of $g$
$g(x)$ must be in the domain of $f$

Composition Quiz 🎯

Composition Practice 🧮

Let $f(x)=2x-1, g(x)=x+5$ .

1) $(f \circ g) (3$ = ?

Composition Concepts 🔽

Exit Quiz ✅

Part 2: Domain of Compositions

🔄 Inverse Functions

Part 2 of 7

What Is an Inverse?

$f^{-1}$ undoes $f$ . If $f(a)=b$ , then .

Part 3: Inverse Functions

🧮 Inverses of Common Functions

Part 3 of 7

Inverse Pairs

Function $f(x)$	Inverse $f^{-1}(x)$	Domain restriction
$x^{2}$

Part 4: Finding Inverses

📊 Composition with Tables & Graphs

Part 4 of 7

Reading from Tables

Given tables of $f$ and $g$ :

$x$	$f(x)$

Part 5: Verifying Inverses

🧩 Piecewise & Absolute Value Compositions

Part 5 of 7

Composing with Piecewise Functions

If $f(x) = \begin{cases} x+2 & x < 0 \\ x^2 & x \geq 0 \end{cases}$ and :

Part 6: Problem-Solving Workshop

📐 Verifying Inverses & Algebraic Techniques

Part 6 of 7

How to Verify Two Functions Are Inverses

If $f$ and $g$ are inverses, BOTH must hold:

$f(g(x)) = x \quad \text{AND} \quad g(f(x)) = x$

Part 7: Review & Applications

🎯 Composition & Inverses — Full Synthesis

Part 7 of 7

Master Summary

Concept	Key Formula
Composition	$(f \circ g)(x) = f(g(x))$
Inverse

$(f \circ g)(x) = 2(x^2-4)+1 = 2x^2-7$

$(g \circ f)(x) = (2x+1)^2-4 = 4x^2+4x-3$

Example 2: Evaluating at a Point

$f(x)=\sqrt{x}, g(x)=3x+1$

$(f \circ g)(5) = f(g(5)) = f(16) = 4$

Example 3: Three Functions

$f(x)=x^2, g(x)=x+1, h(x)=2x$

$(f \circ g \circ h)(x) = f(g(h(x))) = f(g(2x)) = f(2x+1) = (2x+1)^2$

Express $h(x) = \sqrt{3x+7}$ as a composition: let $f(x) = \sqrt{x}, g(x) = 3x+7$ . Then $h = f \circ g$ .

$f(x) = \sqrt{x}, g(x) = 4-x^2$

$(f \circ g)(x) = \sqrt{4-x^2}$

Domain: $4-x^2 \geq 0 \implies -2 \leq x \leq 2$

$f(x) = \frac{1}{x}, g(x) = x-3$

$(f \circ g)(x) = \frac{1}{x-3}$

Domain: $x \neq 3$ (so $g(x) \neq 0$ , which is not in domain of $f$ ).

2) $(g \circ f)(3)$ = ?

3) $(f \circ f)(2)$ = ?

$f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x$

Domain of $f^{-1}$ = Range of $f$
Range of $f^{-1}$ = Domain of $f$
The graph of $f^{-1}$ is the reflection of $f$ across the line $y = x$

When Does $f^{-1}$ Exist?

$f$ must be one-to-one (each output comes from exactly one input).

One-to-one → passes the Horizontal Line Test
Not one-to-one → no inverse (unless we restrict the domain)

📝 Finding Inverse Functions

Algorithm

Replace $f(x)$ with $y$
Swap $x$ and $y$
Solve for $y$
Write $f^{-1}(x) = y$

Example 1: $f(x) = 3x - 7$

$y = 3x-7$ → $x = 3y-7$ → $x+7 = 3y$ →

$f^{-1}(x) = \frac{x+7}{3}$

Verify: $f(f^{-1}(x)) = 3\cdot\frac{x+7}{3}-7 = x+7-7 = x$ ✓

Example 2: $f(x) = x^2+1, x \geq 0$

$x = y^2+1 \implies y = \sqrt{x-1}$

$f^{-1}(x) = \sqrt{x-1}, \quad x \geq 1$

📊 The Horizontal Line Test

Function	One-to-one?	Inverse exists?
$f(x) = 2x+3$	Yes	Yes
$f(x) = x^2$ (all reals)	No	No (without restriction)
$f(x) = x^3$	Yes	Yes ( $f^{-1}(x) = \sqrt[3]{x}$ )
$f(x) = \sin x$ (all reals)	No	No (restrict to $[-\pi/2, \pi/2]$ )
$f(x) = e^x$	Yes	Yes ( $f^{-1}(x) = \ln x$ )

Restricting Domains

$f(x) = x^2$ on $x \geq 0$ : now one-to-one!

$f^{-1}(x) = \sqrt{x}$ (the principal square root)

💡 When we write $\sin^{-1}x$ , we use the restricted domain $[-\pi/2, \pi/2]$ .

Inverse Functions Quiz 🎯

Finding Inverses 🧮

1) $f(x) = 4x - 3$ . Find $f^{-1}(9)$ :

2) $f(x) = \frac{x+1}{2}$ . Find $f^{-1}(x) = ax + b$ . What is $a$ ?

3) Same function: What is $b$ ?

Inverse Concepts 🔽

💡 The graph of each inverse is the reflection of the original function over $y = x$ .

📝 Exponential & Logarithmic Inverses

Why $\ln$ and $e^x$ are inverses

$e^{\ln x} = x \quad (x > 0)$

$\ln(e^x) = x \quad (\text{all } x)$

Solving with Inverses

Solve $e^{2x} = 15$ :

$\ln(e^{2x}) = \ln 15$

$2x = \ln 15$

$x = \frac{\ln 15}{2} \approx 1.354$

Solve $\log_2(x-3) = 5$ :

$x-3 = 2^5 = 32$

$x = 35$

Key Identities

$\log_a(a^x) = x$
$a^{\log_a x} = x$

🔀 Inverses of Rational Functions

Example: $f(x) = \frac{2x+1}{x-3}$

Swap and solve:

$x = \frac{2y+1}{y-3}$

$x(y-3) = 2y+1$

$xy - 3x = 2y + 1$

$xy - 2y = 3x + 1$

$y(x-2) = 3x+1$

$f^{-1}(x) = \frac{3x+1}{x-2}$

Verification: $f(f^{-1}(x))$ :

$f\left(\frac{3x+1}{x-2}\right) = \frac{2 \cdot \frac{3x+1}{x-2}+1}{\frac{3x+1}{x-2}-3} = \frac{\frac{6x+2+x-2}{x-2}}{\frac{3x+1-3x+6}{x-2}} = \frac{7x}{7} = x$ ✓

Inverse Pairs Quiz 🎯

Inverse Calculations 🧮

1) Solve $e^x = 20$ : $x = \ln($ ? $)$ . Enter the number.

2) Solve $\log_3 x = 4$ : $x$ = ?

3) If $f(x) = 5x-3$ , then $f^{-1}(12)$ = ?

Inverse Pairs Concepts 🔽

Find $(f \circ g)(2)$ :

$g(2) = 4$ , then $f(4) = 2$ . So $(f \circ g)(2) = 2$ .

Find $(g \circ f)(3)$ :

$f(3) = 1$ , then $g(1) = 2$ . So $(g \circ f)(3) = 2$ .

📝 Inverse from Tables

If $f$ is one-to-one, we can read $f^{-1}$ from the table by swapping input/output:

$x$	$f(x)$
1	3
2	5
3	1
4	2

So $f^{-1}(3) = 1, f^{-1}(5) = 2, f^{-1}(1) = 3, f^{-1}(2) = 4$ .

Verifying One-to-One from a Table

Check: does any output appear more than once? If yes, $f$ is NOT one-to-one.

Composition Chains from Tables

$(f \circ f)(1)$ : $f(1) = 3$ , then $f(3) = 1$ . So .

This means $1$ and $3$ form a 2-cycle under $f$ .

📈 Composition with Graphs

To find $(f \circ g)(a)$ from graphs:

Go to $x = a$ on the graph of $g$ → read $g(a)$
Go to $x = g(a)$ on the graph of $f$ → read $f(g(a))$

Graph of $f^{-1}$

Reflect the graph of $f$ across $y = x$ .

Key observations:

If $f$ passes through $(2, 5)$ , then $f^{-1}$ passes through $(5, 2)$

Fixed Points

A fixed point is where $f(x) = x$ (the graph crosses $y = x$ ).

At fixed points: $f(a) = a = f^{-1}(a)$ . Both the function and its inverse share this point!

Tables & Graphs Quiz 🎯

Use this table:

$x$	1	2	3	4
$f(x)$	4	1	2	3
$g(x)$	2	3	4	1

Table Practice 🧮

Use: $f(1)=3, f(2)=5, f(3)=7, f(4)=9$

1) $(f \circ f^{-1})(7)$ = ?

2) $f^{-1}(9)$ = ?

3) $f^{-1}(f^{-1}(7))$ = ? (Hint: find $f^{-1}(7)$ first, then apply again)

Graph & Table Concepts 🔽

$(f \circ g)(3) = f(g(3)) = f(2) = 2^2 = 4$ (since $2 \geq 0$ )

$(f \circ g)(-2) = f(g(-2)) = f(-3) = -3+2 = -1$ (since $-3 < 0$ )

Composing with Absolute Value

$|f(x)|$ takes the output and makes it positive.

$f(|x|)$ takes the input and makes it positive first.

These are different! For $f(x) = x - 3$ :

$|f(x)| = |x-3|$ (V-shape at $x=3$ )
$f(|x|) = |x|-3$ (V-shape at $x=0$ , shifted down $3$ )

📝 Function Operations Review

Arithmetic Operations

$(f+g)(x) = f(x)+g(x)$
$(f-g)(x) = f(x)-g(x)$
$(fg)(x) = f(x) \cdot g(x)$
$(f/g)(x) = f(x)/g(x), \quad g(x) \neq 0$

Example

$f(x) = x^2, g(x) = 2x+1$

$(f+g)(x) = x^2+2x+1 = (x+1)^2$

$(fg)(x) = x^2(2x+1) = 2x^3+x^2$

$(f/g)(x) = \frac{x^2}{2x+1}, \quad x \neq -\frac{1}{2}$

Domains of Combined Functions

$\text{dom}(f+g) = \text{dom}(f) \cap \text{dom}(g)$

$\text{dom}(f/g) = \text{dom}(f) \cap \text{dom}(g) \setminus \{x: g(x)=0\}$

🔧 Decomposition Strategies

Breaking a complex function into simpler pieces:

Chain Decomposition (for Calculus)

Complex Function	Inner $g(x)$	Outer $f(u)$
$\sqrt{x^2+1}$	$x^2+1$	$\sqrt{u}$
$(3x-5)^7$	$3x-5$	$u^7$
$\sin(x^2)$	$x^2$	$\sin u$
$e^{-x^2}$	$-x^2$	$e^u$
$\ln(\cos x)$	$\cos x$	$\ln u$

💡 This decomposition is the foundation of the Chain Rule in calculus: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$ .

Operations & Decomposition Quiz 🎯

Operations Practice 🧮

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

1) $(f+g)(4)$ = ?

2) $(f \cdot g)(2)$ = ?

3) $(f/g)(6)$ = ? (Enter as a fraction like "3/4")

Decomposition Concepts 🔽

⚠️ Verifying only ONE direction is not enough! You need both.

Example: Are $f(x) = 3x-6$ and $g(x) = \frac{x+6}{3}$ inverses?

Check 1: $f(g(x)) = 3\cdot\frac{x+6}{3}-6 = x+6-6 = x$ ✓

Check 2: $g(f(x)) = \frac{(3x-6)+6}{3} = \frac{3x}{3} = x$ ✓

Both hold → yes, they are inverses!

🔧 Algebraic Techniques for Finding Inverses

Technique 1: Quadratic Inverses

$f(x) = x^2 - 4x + 7, x \geq 2$

Complete the square: $f(x) = (x-2)^2+3$

Swap: $x = (y-2)^2+3$

$(y-2)^2 = x-3$

$y = 2+\sqrt{x-3}$ (positive root, since $x \geq 2$ )

Technique 2: Implicit Solving

$f(x) = \frac{x^2+1}{x^2-1}, x > 1$

$x = \frac{y^2+1}{y^2-1} \implies x(y^2-1) = y^2+1 \implies xy^2-x = y^2+1$

$y^2(x-1) = x+1 \implies y^2 = \frac{x+1}{x-1} \implies y = \sqrt{\frac{x+1}{x-1}}$

🔗 Composition & Inverse Connections

Self-Inverse Functions (Involutions)

Some functions are their own inverse: $f(f(x)) = x$ .

Examples:

$f(x) = \frac{1}{x}$ : $f(f(x)) = \frac{1}{1/x} = x$ ✓
$f(x) = -x$ : $f(f(x)) = -(-x) = x$ ✓
$f(x) = \frac{a-x}{1+ax}$ for certain $a$

Composition of Inverses

If $h = f \circ g$ , then $h^{-1} = g^{-1} \circ f^{-1}$

💡 The inverse of a composition reverses the order — like undoing layers. Remove the outer layer first!

Derivative Preview

The slope of $f^{-1}$ at a point is the reciprocal of the slope of $f$ :

If $f'(a) = m$ , then $(f^{-1})'(f(a)) = \frac{1}{m}$

Verification & Techniques Quiz 🎯

Verification Practice 🧮

$f(x)=2x+5, g(x)=\frac{x-5}{2}$

1) $f(g(10))$ = ?

2) $g(f(10))$ = ?

3) Are they inverses? (Enter "yes" or "no")

Advanced Inverse Concepts 🔽

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

Essential Inverse Pairs

$x^n \leftrightarrow \sqrt[n]{x}$ , $e^x \leftrightarrow \ln x$ , $a^x \leftrightarrow \log_a x$ , $\sin x \leftrightarrow \sin^{-1}x$ (restricted)

🗺️ Problem-Solving Strategies

Composition

Identify inner and outer functions
Substitute the inner into the outer
Simplify
Check domain restrictions

Finding Inverses

Check one-to-one (HLT or algebraic)
Write $y = f(x)$ , swap $x$ and $y$
Solve for $y$
Verify with $f(f^{-1}(x)) = x$

Decomposition (for Calculus prep)

Identify the "last operation" → outer function
Everything inside → inner function
Practice: $h(x) = e^{\sin(x^2)} \to$ outer $e^u$ , middle , inner

📝 Mixed Practice

Problem 1

$f(x) = \frac{x+1}{x-1}$ . Show $f$ is its own inverse.

$f(f(x)) = \frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1} = \frac{\frac{x+1+x-1}{x-1}}{\frac{x+1-x+1}{x-1}} = \frac{2x}{2} = x$ ✓

Problem 2

$f(x) = 2^x, g(x) = x^2$ . Find $(f \circ g)(3)$ :

$g(3)=9, f(9)=2^9=512$

Problem 3

Find $f^{-1}(x)$ for $f(x) = \ln(x-3)+2$ :

$x = \ln(y-3)+2 \implies x-2 = \ln(y-3) \implies y-3 = e^{x-2}$

$f^{-1}(x) = e^{x-2}+3$

Synthesis Quiz 🎯

Mixed Calculations 🧮

1) $f(x)=3x+1, g(x)=x^2$ . Find $(g \circ f)(-1)$ :

2) $f(x)=\ln x$ . Find $f^{-1}(0)$ :

3) If $f(2) = 7$ and $f(5) = 2$ , find $(f \circ f^{-1})(7)$ :

Master Concepts 🔽

Final Exit Quiz ✅

x−23x+1−32⋅x−23x+1+1=

x−23x+1−3x+6x−26x+2+x−2=

(f \circ f)(1) = 1

Increasing functions have increasing inverses

x

-intercepts of

f

become

y

-intercepts of

f^{-1}

x−1x+1−x+1x−1x+1+x−1=

Function Composition & Inverses

Function Composition & Inverses - Complete Interactive Lesson

Part 1: Function Composition

🔗 Function Composition

What Is Composition?

Example

📝 More Examples

Example 1

🔍 Domain of a Composition

Part 2: Domain of Compositions

🔄 Inverse Functions

What Is an Inverse?

Part 3: Inverse Functions

🧮 Inverses of Common Functions

Inverse Pairs

Part 4: Finding Inverses

📊 Composition with Tables & Graphs

Reading from Tables

Part 5: Verifying Inverses

🧩 Piecewise & Absolute Value Compositions

Composing with Piecewise Functions

Part 6: Problem-Solving Workshop

📐 Verifying Inverses & Algebraic Techniques

How to Verify Two Functions Are Inverses

Part 7: Review & Applications

🎯 Composition & Inverses — Full Synthesis

Master Summary

Example 2: Evaluating at a Point

Example 3: Three Functions

Decomposition

Example

Another Example

Key Properties

When Does f−1f^{-1}f−1 Exist?

📝 Finding Inverse Functions

Algorithm

Example 1: f(x)=3x−7f(x) = 3x - 7f(x)=3x−7

Example 2: f(x)=x2+1,x≥0f(x) = x^2+1, x \geq 0f(x)=x2+1,x≥0

📊 The Horizontal Line Test

Restricting Domains

📝 Exponential & Logarithmic Inverses

Why ln⁡\lnln and exe^xex are inverses

Solving with Inverses

Key Identities

🔀 Inverses of Rational Functions

Example: f(x)=2x+1x−3f(x) = \frac{2x+1}{x-3}f(x)=x−32x+1​

📝 Inverse from Tables

Verifying One-to-One from a Table

Composition Chains from Tables

📈 Composition with Graphs

Graph of f−1f^{-1}f−1

Fixed Points

Composing with Absolute Value

📝 Function Operations Review

Arithmetic Operations

Example

Domains of Combined Functions

🔧 Decomposition Strategies

Chain Decomposition (for Calculus)

Example: Are f(x)=3x−6f(x) = 3x-6f(x)=3x−6 and g(x)=x+63g(x) = \frac{x+6}{3}g(x)=3x+6​ inverses?

🔧 Algebraic Techniques for Finding Inverses

Technique 1: Quadratic Inverses

Technique 2: Implicit Solving

🔗 Composition & Inverse Connections

Self-Inverse Functions (Involutions)

Composition of Inverses

Derivative Preview

Essential Inverse Pairs

🗺️ Problem-Solving Strategies

Composition

Finding Inverses

Decomposition (for Calculus prep)

📝 Mixed Practice

Problem 1

Problem 2

Problem 3

When Does $f^{-1}$ Exist?

Example 1: $f(x) = 3x - 7$

Example 2: $f(x) = x^2+1, x \geq 0$

Why $\ln$ and $e^x$ are inverses

Example: $f(x) = \frac{2x+1}{x-3}$

Graph of $f^{-1}$

Example: Are $f(x) = 3x-6$ and $g(x) = \frac{x+6}{3}$ inverses?