1. 2.6 Operations and Composition

2. Quiz Let f(x) = 3x – 1, g(x) = x2-4, then (f + g)(x) = ______.
(write the function expression )

3. Operations on Functions
Given two functions f and g, suppose Domain of f is Df, Domain of g is Dg
F(x)
Sum
(f + g)(x) = f(x) + g(x)
Domain:
Df intersect Dg
Difference
(f - g)(x) = f(x) – g(x)
Product
(fg)(x) = f(x)g(x)
Quotient
(f/g)(x) = f(x) / g(x)
Df ∪ Dg, where x such that g(x) ≠ 0

4. Operations on Functions
f(x) = 3x -1 g(x) = x2 -4 h(x) = x + 2 p(x) = (x + 3)0.5
Evaluate: (f+g)(6);
Find (fh)(x);
Find (g - f)(x). Is this the same as (f - g)(x)?
Find (g / h)(x)
Find (h - p)(x)
Evaluate: (h - p)(1); (h - p)(-4)

5. Operations on Functions
Discussion
1, h(x) = x2 + 2x
(a) Find h(x + 1)
(a) Find h(x) + h(1)
2, f(x) = 3x + 5
(a) Find f(1/2)
(b) Find f(1) / f(2)

6. Operations on Functions
Example If possible, use the given representations of functions f and g to evaluate (f + g)(4), (f - g)(-2), (fg)(1), and (f/g)(0)
f(x) = 2x g(x) = √ x
Step 1: Find domains of f and g
Step 2: Make sure the input is in the domain of f and g. If it is a quotient operation,
make sure the denominator doesn’t equal to 0.
Step 3: Find corresponding expressions of operations
Step 4: Evaluate new generated functions

7. Difference Quotients
Consider a nonlinear function f and draw a line through two points (x, f(x)) and
(x + h, f(x + h)) y
The slope of the line is: B
y2 – y1
x2 – x1
f(x + h) – f(x)
(x + h) - x = =
f(x + h) – f(x) h A
Difference Quotient x x
x + h

8. Difference Quotient
Example: Find and simplify the difference quotient for the function f given by
f(x) = 3x2 + 2x

9. Homework
PG. 144: 1-4, 12-21(M3), 24, 33, 36, 75, 77, 78
KEY: 24, 36, 78
Reading: 3.2 Quadratic Functions & Graphs