Composite and Inverse Functions

🔗 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

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

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

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

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

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

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

Decomposition

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

🔍 Domain of a Composition

The domain of $f \circ g$ requires:

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

Example

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

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

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

Another Example

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

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

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

Composition Quiz 🎯

Composition Practice 🧮

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

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

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

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

Composition Concepts 🔽

Exit Quiz ✅

🔄 Inverse Functions

Part 2 of 7

What Is an Inverse?

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

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

Key Properties

• 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

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

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

$y = 3x-7$ → $x = 3y-7$ → $x+7 = 3y$ → $y = \frac{x+7}{3}$

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

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

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 🔽

Exit Quiz ✅

🧮 Inverses of Common Functions

Part 3 of 7

Inverse Pairs

💡 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 🔽

Exit Quiz ✅

📊 Composition with Tables & Graphs

Part 4 of 7

Reading from Tables

Given tables of $f$ and $g$:

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:

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 $(f \circ f)(1) = 1$.

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

📈 Composition with Graphs

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

1. Go to $x = a$ on the graph of $g$ → read $g(a)$
2. 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 $(5, 2)$
• Increasing functions have increasing inverses
• $x$-intercepts of $f$ become $y$-intercepts of $f^{-1}$

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:

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 $f^{-1}$ again)

Graph & Table Concepts 🔽

Exit Quiz ✅

🧩 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 $g(x) = x-1$:

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

📝 Transformations Are Compositions

Every transformation from earlier in the course is a composition with a simple linear function.

Outer = vertical, inner = horizontal. That is the same inside/outside rule you used for transformations.

Order Matters

Composition is not commutative. With $f(x) = x^2$ and $g(x) = x + 3$:

• $f(g(x)) = (x + 3)^2$, a horizontal shift left $3$
• $g(f(x)) = x^2 + 3$, a vertical shift up $3$

The Identity Function

The function $i(x) = x$ changes nothing: $f(i(x)) = i(f(x)) = f(x)$. It plays the role for composition that $0$ plays for addition and $1$ plays for multiplication.

🔧 Decomposition Strategies

Breaking a complex function into simpler pieces:

Decomposing a Composite Function

💡 Decomposing a function this way tells you its domain (the inner output must be a legal input for the outer function) and the order for building an inverse: undo the outer layer first.

Transformations & Decomposition Quiz 🎯

Composition Practice 🧮

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

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

2) $g(f(4))$ = ?

3) $g(f(x)) = 2x + k$. What is $k$?

Decomposition Concepts 🔽

Exit Quiz ✅

📐 Verifying Inverses & Algebraic Techniques

Part 6 of 7

How to Verify Two Functions Are Inverses

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

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

⚠️ 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!

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 🔽

Exit Quiz ✅

🎯 Composition & Inverses — Full Synthesis

Part 7 of 7

Master Summary

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

1. Identify inner and outer functions
2. Substitute the inner into the outer
3. Simplify
4. Check domain restrictions

Finding Inverses

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

Decomposition

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

📝 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 ✅