Warm-up for Section 3.2:

3.1B Homework Answers = (-7) 3 = /4 7 = 1/ /5 4 = 1/ /4 4 = 1/ /8 6 = 1/262,       x y ,441x

3.1B Homework Answers Continued… q 3 r  10 11

Operations on Functions Section 3.2B Standard: MM2A5d Essential Question: How do I perform operations with functions?

Vocabulary Power function: a function of the form y = ax b,where a is a real number and b is a rational number Composition: h(x) = g(f(x)) is the composition of a function g with a function f. The domain of h is the set of all x-values such that x is in the domain of f and f(x) is the domain of g.

Investigation 1: New functions can be created from established functions through the operations of addition, subtraction, multiplication, and division. Consider the linear function f(x) = 2x + 1 and the quadratic function g(x) = x 2 – 3. Complete the table below for the selected values of the domain. The first column has been done for you. Table 1: x-203 y = f(x)-3 y = g(x)1

Table 1: x-203 y = f(x)-3 y = g(x)1 Now, keeping the domain fixed, add the range values for f and g to create a new function. Complete the table below to identify the y values for this new function. The first column has been done for you. Table 2: x-203 y = f(x) + g(x)

This new function is denoted y = f(x) + g(x) or y = (f + g)(x). To find the rule for the new function, simply add the expressions for y = f(x) and y = g(x). This new function is: y = (2x + 1) + (x 2 – 3). In simple form, we have y = x 2 + 2x – 2. Let’s call this function h. So, h(x) = x 2 + 2x – 2. Evaluate the function for each domain element to check the values in Table 2.

h(-2) = (-2) 2 + 2(-2) – 2 = _______ h(-1) = (-1) 2 + 2(-1) – 2 = _______ h(0) = (0) 2 + 2(0) – 2 = _______ h(3) = (3) 2 + 2(3) – 2 = _______ Did you get the same values in Table 2? ________ YES

Let’s use f(x) = 2x + 1 and g(x) = x 2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively. (2). f(x) – g(x) or (f – g)(x) = (2x + 1) – (x 2 – 3) = 2x + 1 – x = -x 2 + 2x + 4 s(x) = -x 2 + 2x + 4

Let’s use f(x) = 2x + 1 and g(x) = x 2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively. (3). f(x) ∙ g(x) or (f g)(x) = (2x + 1)(x 2 – 3) = 2x 3 – 6x + x 2 – 3 = 2x 3 + x 2 – 6x – 3 m(x) = 2x 3 + x 2 – 6x – 3

Let’s use f(x) = 2x + 1 and g(x) = x 2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively. (4). or

The domain of the new function is the set of values common to original functions. In other words, it is the intersection of the domains of the original functions. The domain of f(x) = 2x + 1 is _____________ and the domain of g(x) = x 2 – 3 is ___________. So, the domains for h, s, and m will all be ___________. all reals

But, the function d was created by division so we must check to see what values of the common domain will make the denominator zero. This value must be excluded. So, the domain of y = d(x) is all reals except __________. x = -½ 2x + 1 = 0 2x = -1 x = -1/2

Check for Understanding: Let h(x) = 3x + 1 and p(x) = 2x – 5 Find the following and state the domain. (5). h(x) + p(x) or (h + p)(x) = ____________ Domain: _____________ (6). h(x) – p(x) or (h – p)(x) = ___________ Domain: _____________ 5x – 4 all reals x + 6 all reals

Check for Understanding: Let h(x) = 3x + 1 and p(x) = 2x – 5 Find the following and state the domain. (7). h(x) ∙ p(x) or (hp)(x) = ____________ Domain: _____________ (8). or = ___________ Domain: ______________________ 6x 2 – 13x – 5 all reals all reals except x = 5/2

Another way of combining two functions is to form the composition of one with the other DfDf DgDg RfRf RgRg f(x) = x + 1 g(x) = x 2 The composition of g with f can be pictured above.

The new function created maps the domain of f to the range of g DfDf DgDg RfRf RgRg f(x) = x + 1 g(x) = x 2 If we call this new function h, then the rule for h is h(x) = (x + 1) 2

The domain of h is the set of all x-values such That x is in the domain of g and g(x) is in the domain of f. h(x) = f(g(x))

( 9). Let f(x) = 6x and g(x) = 3x + 5 find each composition and its domain. a. f(g(x)) = Domain: _________ b. g(f(x)) = Domain: _________ f(3x + 5) = 6(3x + 5) g(6x)g(6x) = 3(6x) + 5 h(x) = 18x + 5 all reals h(x) = 18x + 5 all reals

( 9). Let f(x) = 6x and g(x) = 3x + 5 find each composition and its domain. c. f(f(x)) = Domain: __________ d. g(g(x)) = Domain: _________ g(3x + 5) = 3(3x + 5) + 5 h(x) = 9x + 20 = 9x f(6x)f(6x) = 6(6x) h(x) = 36x all reals

(10). Let f(x) = 2x and g(x) = x 2 – 3 find each composition and its domain. a. f(g(x)) = Domain = __________ b. g(f(x) = Domian = _________ f(x 2 – 3 ) = 2(x 2 – 3 ) h(x) = 2x 2 – 6 g(2x )g(2x ) = (2x) 2 – 3 h(x) = 4x 2 – 3 all reals

(10). Let f(x) = 2x and g(x) = x 2 – 3 find each composition and its domain. c. g(g(x)) = Domian = _________ g(x 2 – 3 ) = (x 2 – 3) 2 – 3 h(x) = x 4 – 6x = (x 2 – 3)(x 2 – 3) – 3 = x 4 – 3x 2 – 3x – 3 all reals

Check for Understanding: Let p(x) = 3x + 1 and h(x) = x 2 – 4, find each new functions and its domain. (11). (p + h)(x) = _____________________ Domain: ________________ (12). (h – p)(x) = _____________________ Domain: ________________ (13). (ph)(x) = _____________________ Domain: ________________ x 2 + 3x – 3 x 2 – 3x – 5 3x 3 + x 2 – 12x – 4 all reals

(14). = _____________________ Domain: ___________________ (15). p(h(x)) = _____________________ Domain: ________________ (16). h(p(x)) = _____________________ Domain: ________________ all reals all reals except x = ±2 3x 2 – 11 9x 2 + 6x – 3