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Part 1: Function Composition
๐ Function Composition
Part 1 of 7
What Is Composition?
The composition of f and g, written (fโg)(x), means "f of g of x":
(fโg)(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)=x2 and g(x)=x+3:
(fโg)(x)=f(g(x))=f(x+3)
(gโf)(x)=g(f(x))=g(x2
โ ๏ธ Order matters! fโg๎ =gโf in general.
๐ More Examples
Example 1
f(x)=2x+1,g(x)=x2โ4
๐ Domain of a Composition
The domain of fโg requires:
- x must be in the domain of g
- g(x) must be in the domain of
Composition Practice ๐งฎ
Let f(x)=2xโ1,g(x)=x+5.
1) = ?
Composition Concepts ๐ฝ
Part 2: Domain of Compositions
๐ Inverse Functions
Part 2 of 7
What Is an Inverse?
fโ1 undoes f. If f(a)=b, then .
Part 3: Inverse Functions
๐งฎ Inverses of Common Functions
Part 3 of 7
Inverse Pairs
| Function f(x) | Inverse fโ1(x) | Domain restriction |
|---|
|
Part 4: Finding Inverses
๐ Composition with Tables & Graphs
Part 4 of 7
Reading from Tables
Given tables of f and g:
Part 5: Verifying Inverses
๐งฉ Piecewise & Absolute Value Compositions
Part 5 of 7
Composing with Piecewise Functions
If f(x)={x+2 and :
Part 6: Problem-Solving Workshop
๐ 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
Part 7: Review & Applications
๐ฏ Composition & Inverses โ Full Synthesis
Part 7 of 7
Master Summary
| Concept | Key Formula |
|---|
| Composition | (fโg)(x)=f(g(x)) |
| Inverse | |
=
(x+
3)2
)
=
x2+
3
(fโg)(x)=2(x2โ4)+1=2x2โ7
(gโf)(x)=(2x+1)2โ4=4x2+4xโ3
Example 2: Evaluating at a Point
f(x)=xโ,g(x)=3x+1
(fโg)(5)=f(g(5))=f(16)=4
Example 3: Three Functions
f(x)=x2,g(x)=x+1,h(x)=2x
(fโgโh)(x)=f(g(h(x)))=f(g(2x))=f(2x+1)=(2x+1)2
Decomposition
Express h(x)=3x+7โ as a composition: let f(x)=xโ,g(x)=3x+7. Then h=fโg.
f
Example
f(x)=xโ,g(x)=4โx2
(fโg)(x)=4โx2โ
Domain: 4โx2โฅ0โนโ2โคxโค2
Another Example
f(x)=x1โ,g(x)=xโ3
(fโg)(x)=xโ31โ
Domain: x๎ =3 (so g(x)๎ =0, which is not in domain of f).
(fโg)(3)
2) (gโf)(3) = ?
3) (fโf)(2) = ?
fโ1(b)=a
f(fโ1(x))=xandfโ1(f(x))=x
Key Properties
- Domain of fโ1 = Range of f
- Range of fโ1 = Domain of f
- The graph of fโ1 is the reflection of f across the line y=x
When Does fโ1 Exist?
f must be one-to-one (each output comes from exactly one input).
- One-to-one โ passes the Horizontal Line Test
- Not one-to-one โ no inverse (unless we restrict the domain)
๐ Finding Inverse Functions
Algorithm
- Replace f(x) with y
- Swap x and y
- Solve for y
- Write fโ1(x)=y
Example 1: f(x)=3xโ7
y=3xโ7 โ x=3yโ7 โ x+ โ
fโ1(x)=3x+7โ
Verify: f(fโ1(x))=3โ
3 โ
Example 2: f(x)=x2+1,xโฅ0
x=y2+1โนy=xโ1
fโ1(x)=xโ1
๐ The Horizontal Line Test
| Function | One-to-one? | Inverse exists? |
|---|
| f(x)=2x+3 | Yes | Yes |
| f(x)=x2 (all reals) | No | No (without restriction) |
| f(x)=x3 | Yes | Yes (fโ1(x)=) |
| f(x)=sinx (all reals) | No | No (restrict to [โฯ/2,ฯ/2]) |
| f(x)=ex | Yes | Yes (fโ1(x)=) |
Restricting Domains
f(x)=x2 on xโฅ0: now one-to-one!
fโ1(x)=xโ (the principal square root)
๐ก When we write sinโ1x, we use the restricted domain [โฯ/2,ฯ/2].
Inverse Functions Quiz ๐ฏ
Finding Inverses ๐งฎ
1) f(x)=4xโ3. Find fโ1(9):
2) f(x)=2x+1โ. Find fโ1(x)=ax+b. What is a?
3) Same function: What is b?
x2
| xโฅ0 for both |
| x3 | 3xโ | All reals |
| ex | lnx | x>0 for ln |
| 10x | log10โx | x>0 for log |
| ax | logaโx | x>0 for logaโ |
| sinx | sinโ1x | [โฯ/2,ฯ/2], [โ1,1] |
| cosx | cosโ1x | [0,ฯ], [โ1,1] |
| tanx | tanโ1x | (โฯ/2,ฯ/2) |
๐ก The graph of each inverse is the reflection of the original function over y=x.
๐ Exponential & Logarithmic Inverses
Why ln and ex are inverses
elnx=x(x>0)
ln(ex)=x(allย x)
Solving with Inverses
Solve e2x=15:
ln(e2x)=ln15
2x=ln15
x=2ln15โโ1.354
Solve log2โ(xโ3)=5:
xโ3=25=32
x=35
Key Identities
- logaโ(ax)=x
- a
๐ Inverses of Rational Functions
Example: f(x)=xโ32x+1โ
Swap and solve:
x=yโ32y+1โ
x(yโ3)=2y+1
xyโ3x=2y+1
xyโ2y=3x+1
y(xโ2)=3x+1
fโ1(x)=xโ23x+1โ
Verification: f(fโ1(x)):
f(xโ โ
Inverse Calculations ๐งฎ
1) Solve ex=20: x=ln(?). Enter the number.
2) Solve log3โx=4: x = ?
3) If f(x)=5xโ3, then fโ1(12) = ?
Inverse Pairs Concepts ๐ฝ
g(x)
Find (fโg)(2):
g(2)=4, then f(4)=2. So (fโg)(2)=2.
Find (gโf)(3):
f(3)=1, then g(1)=2. So (gโf)(3)=2.
๐ Inverse from Tables
If f is one-to-one, we can read fโ1 from the table by swapping input/output:
So fโ1(3)=1,fโ1.
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
(fโf)(1): f(1)=3, then f(3)=. So .
This means 1 and 3 form a 2-cycle under f.
๐ Composition with Graphs
To find (fโg)(a) from graphs:
- Go to x=a on the graph of g โ read g(a)
- Go to x=g(a) on the graph of f โ read f(g(a))
Graph of fโ1
Reflect the graph of f across y=x.
Key observations:
- If f passes through (2,5), then fโ1 passes through (5,
Fixed Points
A fixed point is where f(x)=x (the graph crosses y=x).
At fixed points: f(a)=a=fโ1(a). Both the function and its inverse share this point!
Tables & Graphs Quiz ๐ฏ
Use this table:
| x | 1 | 2 | 3 | 4 |
|---|
| f(x) | 4 | 1 | 2 | 3 |
| g(x) | 2 | 3 | 4 | 1 |
Table Practice ๐งฎ
Use: f(1)=3,f(2)=5,f(3)=7,f(4)=9
1) (fโfโ1)(7) = ?
2) fโ1(9) = ?
3) fโ1(fโ1(7)) = ? (Hint: find f first, then apply again)
Graph & Table Concepts ๐ฝ
x
2
โ
x<0xโฅ0โ
g(x)=xโ1 (fโg)(3)=f(g(3))=f(2)=22=4 (since 2โฅ0)
(fโg)(โ2)=f(g(โ2))=f(โ3)=โ3+2=โ1 (since โ3<0)
Composing with Absolute Value
โฃf(x)โฃ takes the output and makes it positive.
f(โฃxโฃ) takes the input and makes it positive first.
These are different! For f(x)=xโ3:
- โฃf(x)โฃ=โฃxโ3โฃ (V-shape at x=3)
- f(โฃxโฃ)=โฃxโฃโ3 (V-shape at x=0, shifted down 3)
๐ Function Operations Review
Arithmetic Operations
- (f+g)(x)=f(x)+g(x)
- (fโg)(x)=f(x)โg(x)
- (fg)(x)=f(x)โ
g(x)
- (f/g)(x)=f(x)/g(x),g(x)๎ =
Example
f(x)=x2,g(x)=2x+1
(f+g)(x)=x2+2x+1=
(fg)(x)=x2(2x+1)=2x
(f/g)(x)=2x+1x2
Domains of Combined Functions
dom(f+g)=dom(f)โฉdom(g)
dom(f/g)=dom(f)โฉdom(g)โ{x:g(
๐ง Decomposition Strategies
Breaking a complex function into simpler pieces:
Chain Decomposition (for Calculus)
| Complex Function | Inner g(x) | Outer f(u) |
|---|
| x2+1โ | x2+1 | uโ |
| (3xโ5)7 | 3xโ5 | u |
| sin(x2) | x2 | sinu |
| eโx2 | โx2 | |
| ln(cosx) | cosx | lnu |
๐ก This decomposition is the foundation of the Chain Rule in calculus: dxdโf(g(x))=.
Operations & Decomposition Quiz ๐ฏ
Operations Practice ๐งฎ
f(x)=x+3,g(x)=2x
1) (f+g)(4) = ?
2) (fโ
g)(2) = ?
3) (f/g)(6) = ? (Enter as a fraction like "3/4")
Decomposition Concepts ๐ฝ
AND
g
(
f
(
x
))
=
x
โ ๏ธ Verifying only ONE direction is not enough! You need both.
Example: Are f(x)=3xโ6 and g(x)=3x+6โ inverses?
Check 1: f(g(x))=3โ
3x+6โโ6=x+6โ6=x โ
Check 2: g(f(x))=3(3xโ6)+6โ=33xโ=x โ
Both hold โ yes, they are inverses!
๐ง Algebraic Techniques for Finding Inverses
Technique 1: Quadratic Inverses
f(x)=x2โ4x+7,xโฅ2
Complete the square: f(x)=(xโ2)2+3
Swap: x=(yโ2)2+3
(yโ2)2=xโ3
y=2+xโ3โ (positive root, since )
Technique 2: Implicit Solving
f(x)=x2โ1x
x=y2โ1
y2(xโ1)=x+1โน
๐ Composition & Inverse Connections
Self-Inverse Functions (Involutions)
Some functions are their own inverse: f(f(x))=x.
Examples:
- f(x)=x1โ: f(f(x))=1/x1โ=x โ
- f(x)=โx: f(f(x))=โ(โx)=x โ
- f(x)=1+axaโxโ for certain a
Composition of Inverses
If h=fโg, then hโ1=g
๐ก The inverse of a composition reverses the order โ like undoing layers. Remove the outer layer first!
Derivative Preview
The slope of fโ1 at a point is the reciprocal of the slope of f:
If fโฒ(a)=m, then (fโ1
Verification & Techniques Quiz ๐ฏ
Verification Practice ๐งฎ
f(x)=2x+5,g(x)=2xโ5โ
1) f(g(10)) = ?
2) g(f(10)) = ?
3) Are they inverses? (Enter "yes" or "no")
Advanced Inverse Concepts ๐ฝ
f(fโ1(x))=fโ1(f(x))=x
| Finding inverse | Swap x,y and solve |
| One-to-one test | Horizontal Line Test |
| Self-inverse | f(f(x))=x |
| Inverse of composition | (fโg)โ1=gโ1โfโ1 |
Essential Inverse Pairs
xnโnxโ, exโlnx, axโlogaโx, sinxโsinโ1x (restricted)
๐บ๏ธ 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(fโ1(x))=x
Decomposition (for Calculus prep)
- Identify the "last operation" โ outer function
- Everything inside โ inner function
- Practice: h(x)=esin(x2)โ outer , middle , inner
๐ Mixed Practice
Problem 1
f(x)=xโ1x+1โ. Show f is its own inverse.
f(f(x))= โ
Problem 2
f(x)=2x,g(x)=x2. Find :
g(3)=9,f(9)=29=512
Problem 3
Find fโ1(x) for f(x)=ln(xโ3)+:
x=ln(yโ3)+2โนxโ2=
fโ1(x)=exโ2+3
Mixed Calculations ๐งฎ
1) f(x)=3x+1,g(x)=x2. Find (gโf)(โ1):
2) f(x)=lnx. Find fโ1(0):
3) If f(2)=7 and f(5)=2, find (fโ:
7=
3y
y=3x+7โ x+7
โ
โ
7=
x+
7โ
7=
x
โ
โ
,
x
โฅ
1
lnx
logaโx
=
x
2
3x+1
โ
)
=
xโ23x+1โโ32โ
xโ23x+1โ+1โ=
xโ23x+1โ3x+6โxโ26x+2+xโ2โโ=
77xโ=
x
(
5
)
=
2,fโ1(1)=
3,fโ1(2)=
4
1
(fโf)(1)=1
2
)
Increasing functions have increasing inversesx-intercepts of f become y-intercepts of fโ1 โ1
(
7
)
0
(
x
+
1)2
3
+
x2
โ
,
x
๎ =
โ21โ
x
)
=
0}
7
e
u
fโฒ
(
g
(
x
))
โ
gโฒ(x)
x
โฅ
2
2
+
1
โ
,
x
>
1
y2
+
1
โ
โน
x(y2โ
1)=
y2+
1โน
xy2โ
x=
y2+
1
y2=
xโ1x+1โโน
y=
โ1
โ
fโ1
)โฒ
(
f
(
a
))
=
m1โ
e
u
xโ1x+1โ
โ
1
xโ1x+1โ+1
โ
=
xโ1x+1โx+1โxโ1x+1+xโ1โโ=
22xโ=
x
(fโg)(3)
2
ln(yโ
3)โน
yโ
3=
exโ2
fโ1
)
(
7
)