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9-4

Operations with Functions. 9-4. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra2. x – 3. x – 2. Warm Up Simplify. Assume that all expressions are defined. 1. (4 x + 5) – ( x 2 + 3 x – 4). – x 2 + x + 9. 3. Objectives. Write and evaluate composite functions.

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9-4

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  1. Operations with Functions 9-4 Warm Up Lesson Presentation Lesson Quiz Holt Algebra2

  2. x – 3 x – 2 Warm Up Simplify. Assume that all expressions are defined. 1. (4x + 5) – (x2 + 3x – 4) –x2 + x + 9 3.

  3. Objectives Write and evaluate composite functions. Alg 2 Standard: 24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.

  4. Vocabulary composition of functions

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

  6. 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.

  7. Caution! Be careful not to confuse the notation for multiplication of functions with composition fg(x) ≠ f(g(x))

  8. Example 3A: Evaluating Composite Functions Given f(x) = 2xand g(x) = 7 – x, find each value. 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.

  9. Check It Out! Example 3a Given f(x) = 2x – 3 and g(x) = x2, find each value. 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.

  10. Check It Out! Example 3b Given f(x) = 2x –3and g(x) = x2, find each value. 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.

  11. 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.

  12. Given f(x) = x2–1and g(x) = , write each composite function. State the domain of each. x 1 – x f(g(x)) = f( ) x x = = ()2– 1 1 – x 1 – x –1 + 2x (1 – x)2 Example 4A: Writing Composite Functions f(g(x)) Substitute the rule g into f. Use the rule for f. Note that x ≠ 1. Simplify. The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because g(1) is undefined.

  13. Given f(x) = x2–1and g(x) = , write each composite function. State the domain of each. (x2 – 1) x = 1 – (x2 – 1) 1 – x = x2– 1 2 – x2 Example 4B: Writing Composite Functions g(f(x)) g(f(x)) = g(x2 – 1) Substitute the rule f into g. Use the rule for g. Simplify. Note that x ≠ . The domain of g(f(x)) is x ≠ or {x|x ≠ } because f( ) = 1 and g(1) is undefined.

  14. Given f(x) = 3x –4and g(x) = + 2 , write each composite. State the domain of each. f(g(x)) = 3( + 2) – 4 = + 6 – 4 = + 2 Check It Out! Example 4a f(g(x)) Substitute the rule g into f. Distribute. Note that x ≥ 0. Simplify. The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.

  15. Given f(x) = 3x –4and g(x) = + 2 , write each composite. State the domain of each. g(f(x)) = 4 4 4 3 3 3 = Note that x ≥ . The domain of g(f(x)) is x ≥ or {x|x ≥ }. Check It Out! Example 4b g(f(x)) Substitute the rule f into g.

  16. Given f(x) = x2and g(x) = , write each composite function. State the domain of each. Lesson Quiz f(g(x)) = x – 1; 5. f(g(x)) {x|x ≥ 1} 6. g(f(x)) {x|x ≤ – 1 or x ≥ 1} Given f(x) = 4x2– 1and g(x) = 2x – 1, find each function or value. 4. g(f(2)) 29

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