Operations with Functions Essential Questions How do we add, subtract, multiply, and divide functions? How do we write and evaluate composite functions? Holt McDougal Algebra 2 Holt Algebra2

Another function operation uses the output from one function as the input for a second function. This operation is called the composition of functions.

The order of function operations is the same as the order of operations for numbers and expressions. To find f(g(3)), evaluate g(3) first and then substitute the result into f.

The composition (f o g)(x) or f(g(x)) is read “f of g of x.” Reading Math Be careful not to confuse the notation for multiplication of functions with composition fg(x) ≠ (f o g)(x) Caution!

Evaluating Composite Functions Given f(x) = 2x and g(x) = 7 – x, find each value. 1. f(g(4)) Step 1 Find g(4) g(4) = 7 – 4 g(x) = 7 – x = 3 Step 2 Find f (3) f (3) = 23 f(x) = 2x = 8 So f(g(4)) = 8.

Evaluating Composite Functions Given f(x) = 2x and g(x) = 7 – x, find each value. 2. g(f(4)) Step 1 Find f (4) f(4) = 24 f(x) = 2x = 16 Step 2 Find g(16) g(16) = 7 – 16 g(x) = 7 – x. = –9 So g(f (4)) = –9.

Evaluating Composite Functions Given f(x) = 2x – 3 and g(x) = x2, find each value. 3. f(g(3)) Step 1 Find g(3) g(3) = 32 g(x) = x2 = 9 Step 2 Find f (9) f(9) = 2(9) – 3 f(x) = 2x – 3 = 15 So f(g(3)) = 15.

Evaluating Composite Functions Given f(x) = 2x – 3 and g(x) = x2, find each value. 4. g(f(3)) Step 1 Find f (3) f(3) = 2(3) – 3 f(x) = 2x – 3 = 3 Step 2 Find g(3) g(3) = 32 g(x) = x2 = 9 So g(f (3)) = 9.

You can use algebraic expressions as well as numbers as inputs into functions. To find a rule for f(g(x)), substitute the rule for g into f.

Writing Composite Functions Given f(x) = x2 – 1 and g(x) = , write each composite function. State the domain of each. x 1 – x 5. f(g(x)) f (g(x)) = f ( ) x 1 – x Substitute the rule g into f. = ( )2 – 1 x 1 – x Use the rule for f. Note that x ≠ 1. –1 + 2x (1 – x)2 = Simplify. The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because g(1) is undefined.

Writing Composite Functions Given f(x) = x2 – 1 and g(x) = , write each composite function. State the domain of each. x 1 – x 6. g(f (x)) g (f (x)) = g (x2 – 1) Substitute the rule f into g. (x2 – 1) 1 – (x2 – 1) = Use the rule for g. = x2 – 1 2 – x2 Simplify. The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because g(1) is undefined.

Writing Composite Functions Given f(x) = 3x – 4 and g(x) = + 2 , write each composite. State the domain of each. 7. f(g(x)) Substitute the rule g into f. f(g(x)) = 3( + 2) – 4 = + 6 – 4 Distribute. Note that x ≥ 0. = + 2 Simplify. The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.

Writing Composite Functions Given f(x) = 3x – 4 and g(x) = + 2 , write each composite. State the domain of each. 8. g(f (x)) g(f(x)) = Substitute the rule f into g. Note that x ≥ . 4 3 = The domain of g(f(x)) is x ≥ or {x|x ≥ }. 4 3

Lesson 14.1 Practice B