๐Ÿƒ Flashcards ยท 2.7

Composition
of Functions

12 cards ยท 2 print sheets. Print double-sided, cut along dashed lines.

๐Ÿ“‹ Notes ๐Ÿƒ Flashcards ๐Ÿง  Quiz
Print double-sided (flip on long edge) then cut along dashed lines.
12 cards ยท 2 print sheets
๐Ÿ“„ Page 1 โ€” Questions FRONT ยท Sheet 1/2
1/12
2.7 ยท Composite Notation
What do (f โˆ˜ g)(x) and f(g(x)) mean? How are they related?
2/12
2.7 ยท Order of Operations
To evaluate f(g(x)), which function do you evaluate FIRST?
Inside-out or outside-in?
3/12
2.7 ยท Apply: Numeric
f(x) = 3xโˆ’5 and g(x) = 2x+1. Evaluate f(g(3)).
Step 1: g(3) = ?   Step 2: f(?) = ?
4/12
2.7 ยท Apply: Numeric
f(x) = 3xโˆ’5 and g(x) = 2x+1. Evaluate g(f(3)).
Step 1: f(3) = ?   Step 2: g(?) = ?
5/12
2.7 ยท Commutative?
Is f(g(x)) always equal to g(f(x))? Explain.
Think about what changes when you switch the order
6/12
2.7 ยท Apply: Expression
f(x) = 3xโˆ’5 and g(x) = 2x+1. Write an expression for f(g(x)).
Substitute g(x) = 2x+1 into f
7/12
2.7 ยท Apply: Expression
f(x) = 3xโˆ’5 and g(x) = 2x+1. Write an expression for g(f(x)).
Substitute f(x) = 3xโˆ’5 into g
8/12
2.7 ยท Apply: Equations
h(x) = 2xโˆ’3 and k(x) = xยฒ+4x+5. Evaluate k(h(2)).
h(2) = ?  โ†’  k(?) = ?
๐Ÿ“„ Page 2 โ€” Answers BACK ยท columns swapped
2/12
โœ“ Order of Operations
Evaluate g FIRST (the inner / right function), then plug into f
Work inside-out. f(g(x)): evaluate g(x) first, use result as input for f. "f of g of x" โ†’ g runs first.
1/12
โœ“ Composite Notation
(f โˆ˜ g)(x) = f(g(x)) โ€” they are identical notation
Both mean: evaluate g(x) first, then pass that output as input to f. Read as "f of g of x."
4/12
โœ“ Apply: g(f(3)) = 9
f(3) = 3(3)โˆ’5 = 4  โ†’  g(4) = 2(4)+1 = 9
Note: f(g(3)) = 16 but g(f(3)) = 9. Different order โ†’ different answer!
3/12
โœ“ Apply: f(g(3)) = 16
g(3) = 2(3)+1 = 7  โ†’  f(7) = 3(7)โˆ’5 = 16
Always start with the inner function. g runs first because it's the argument of f.
6/12
โœ“ Apply: f(g(x)) = 6xโˆ’2
f(2x+1) = 3(2x+1)โˆ’5 = 6x+3โˆ’5 = 6xโˆ’2
Substitute g(x) = 2x+1 wherever x appears in f(x) = 3xโˆ’5, then simplify.
5/12
โœ“ Commutative?
NO โ€” composition is NOT commutative in general
f(g(x)) = 6xโˆ’2 but g(f(x)) = 6xโˆ’9. The results are different. Order matters!
8/12
โœ“ Apply: k(h(2)) = 10
h(2) = 2(2)โˆ’3 = 1  โ†’  k(1) = 1+4+5 = 10
k(x) = xยฒ+4x+5. k(1) = 1ยฒ+4(1)+5 = 1+4+5 = 10.
7/12
โœ“ Apply: g(f(x)) = 6xโˆ’9
g(3xโˆ’5) = 2(3xโˆ’5)+1 = 6xโˆ’10+1 = 6xโˆ’9
Substitute f(x) = 3xโˆ’5 into g(x) = 2x+1. Compare: f(g(x)) = 6xโˆ’2 โ‰  6xโˆ’9.
๐Ÿ“„ Page 3 โ€” Questions FRONT ยท Sheet 2/2
9/12
2.7 ยท Apply: Radical
k(x) = xยฒ+4x+5 and p(x) = โˆš(3x+1). Write an expression for (k โˆ˜ p)(x).
Substitute p(x) into k
10/12
2.7 ยท Apply: Table
Using the table below, evaluate (g โˆ˜ h)(1) where h(1) = โˆ’4.
x: โˆ’4, โˆ’1, 2, 5, 7  ยท  g(x): ฯ€, 3, 3, โˆ’2, 4
11/12
2.7 ยท Not Defined
When is f(g(x)) NOT defined?
Think about domain issues
12/12
2.7 ยท Triple Composition
k(x) = (xโˆ’2)ยฒโˆ’3. h(x) = |โˆ’7+3x| for x<1 and โˆ’x for x>1. f(13) = undefined. Evaluate (gโˆ˜fโˆ˜hโˆ˜k)(3).
Work right to left: k first, then h, then f
๐Ÿ“„ Page 4 โ€” Answers BACK ยท columns swapped
10/12
โœ“ Apply: Table
h(1) = โˆ’4  โ†’  g(โˆ’4) = ฯ€
Step 1: h(1) = โˆ’4 (given). Step 2: Look up g(โˆ’4) in the table = ฯ€. So (gโˆ˜h)(1) = ฯ€.
9/12
โœ“ Apply: Radical
3x + 6 + 4โˆš(3x+1)
k(p(x)) = (โˆš(3x+1))ยฒ + 4โˆš(3x+1) + 5 = (3x+1) + 4โˆš(3x+1) + 5 = 3x+6+4โˆš(3x+1).
12/12
โœ“ Triple Composition
Not defined
k(3)=โˆ’2 โ†’ h(โˆ’2)=|โˆ’7+3(โˆ’2)|=13 โ†’ f(13)=not defined. Once any step is undefined, the whole composition is undefined.
11/12
โœ“ Not Defined
When any intermediate output is outside the domain of the next function
If g(a) produces a value not in the domain of f, then f(g(a)) is not defined. Always check each step.