L4.2 – Operations on Functions

1. L4.2 – Operations on Functions Creating new functions by combining functions: 1) Arithmetically 2) Using Composition What is the domain of the new function?

2. Operations on Functions: Arithmetic (1/4) • Functions combined arithmetically create a new function. SUM(f + g)(x) = f(x) + g(x) DIFFERENCE(f – g)(x) = f(x) – g(x) PRODUCT (f . g)(x) = f(x) . g(x) QUOTIENT • The domain of the new function is the intersection of the domains of the original functions. Arithmetic Combinations Operate in Parallel f f(x) x + f(x) + g(x) g x g(x) Any domain restrictions of either f or g, apply to their arithmetic combination. Notice that the outputs (values of the functions) are arithmetically combined. (f + g)(x) is two functions running in parallel on the sameinput. vs. f(a + b) is one function with two inputs (combined prior to running the function)

3. Arithmetic Operations on Functions: (2/4) (A) 3x2 + 5x – 8 Arithmetic operations can be performed • By creating the new function and then evaluating Ex 1: h(x) = 3x2 + 4x, j(x) = x – 8. Find (h + j)(x) = Then evaluate: (h + j)(2) = 3(2)2 + 5(2) − 8 = 14 • Numerically, by combining the output values of the orig’l fcns. From Ex 1 above: (h +j)(2) = Find (f + g)(–1) (f – g)(2) (f .g)(2) For what values of x is (f – g)(x) > 0? h(2) +j(2)= 20 +(-6)= 14 Ex 2: = f(–1) + g(–1) = 7 + 1 = 8 = f(2) – g(2) = 4 – 4 = 0 = f(2) ∙ g(2) = 4.4 =16 f is abovegfor {x| –2 < x < 2}

4. Arithmetic Operations on Functions (3/4) (A) • The difference function (f – g)(x) can be used to solve the equation f(x) = g(x), because f(x) = g(x) f(x) – g(x) = 0. • So to find intersections between f(x) = 2x2 and g(x) = x + 1,you can • Graph the two functions independentlyand use [F5]Math → Intersection Or • Graph the difference function (f – g)(x) = 2x2 – (x + 1) and use [F5]Math → Zero • Either way you get x = −0.5, x = 1 and you can find corresponding y values (1, 2) (−.5, .5) (−.5, 0) (1, 0)

5. 4x2 8x3 – 4x2 2x – 1, x  0 Domain is all Reals Operations on Functions: Arithmetic (4/4) (A) Let f(x) = 4x2 – 2x, g(x) = 2x, h(x) = , j(x) = Find each new function and state its’ domain. 1. (f + g)(x) 2. (f . g)(x) 3. 4. (j – h)(x) Domains: f: all R g: all R h: j: Ready for the answers? Hint: Recall that the domain of the new function is the intersection of the domains of the original functions. So what are the domains of the original functions?

6. Operations on Functions: Composition (1/3) • Functions can be combined using composition. The output of one function is sent as input into the other. (f og)(x) = f(g(x)) x g(x) f(g(x)) • Note that (f og)(x) is not the same as (g of)(x). • The domain of (f og)(x) is the subset of the domain of g which produces output which is in the domain of f. In other words: All x in Domain of gandg(x) in Domain of f Composition Operates in Sequence f g g f x (f ▫ g)(x) f o g g(x)

7. x2 – 7x + 10, , since this becomes input to f(x). D of f og is Operations on Functions: Composition (2/3) (A) • Let f(x) = x4 – 3x2 and g(x) = Find (f og)(x) and give the domain of the composite. 2. Find the domain of (f og)(x), if and Exclude x = 1 from domain of (f og)(x), since g(x) runs 1st. . Also exclude x = -1.

8. Operations on Functions: Composition (3/3) • Sometimes, (f og)(x) = (g of)(x). For ex, let f(x) = 2x + 6 and g(x) = Show that (f og)(x) = (g of)(x) for all x. • You can also evaluate numerically, ≈ for arithmetic combos. Let f(x) = x + 2 and g(x) = x2 + x – 1. Find f(g(3)) and f(g(x)). Find f(f(4)) and f(f(x)). • More complex functions can be rewritten as compositions. Let f(x) = x3, g(x) = , and h(x) = x – 4, and j(x) = 2x. Rewrite k(x) = and l(x) = (2x – 4)3 as compositions of the above functions. Do on board… f(g(3)) = f(11) = 13; f(g(x)) = x2 + x + 1 f(f(4)) = f(6) = 8; f(f(x)) = x + 4 k(x) = g(f(h(x)) l(x) = f(h(j(x))

9. WARMUP (A) Let and Evaluate each expression. Give answers in simplest form. 1. 2. 3. 4. 5. 6. Ready for answers?

10. L4.1 Homework Questions?