Information Technology Mathematics (1) Section 3 2024-2025 PDF

Year: 2024-2025 Fall Semester Mathematics (1) Section 3 Outline Operations of Functions Composite Functions 2  Operations of Functions 3 Example 1: 𝑓 𝑥 = 2𝑥 + 3 , 𝑔 𝑥 = 𝑥 2 − 1 Find and simplify expressions for the following functions. In addition, find t...

Year: 2024-2025 Fall Semester Mathematics (1) Section 3 Outline Operations of Functions Composite Functions 2  Operations of Functions 3 Example 1: 𝑓 𝑥 = 2𝑥 + 3 , 𝑔 𝑥 = 𝑥 2 − 1 Find and simplify expressions for the following functions. In addition, find the domain of each of these functions. 1. 𝑓 + 𝑔 𝑥 2. 𝑓 − 𝑔 𝑥 3. 𝑓𝑔 𝑥 4. 𝑓/𝑔 𝑥 Solution 4 Example 1: 𝑓 𝑥 = 2𝑥 + 3 , 𝑔 𝑥 = 𝑥2 − 1 Solution 𝑓 + 𝑔 𝑥 = 2𝑥 + 3 + (𝑥 2 −1) = 𝑥 2 + 2𝑥 + 2 , and the domain is (−∞, ∞). 𝑓 − 𝑔 𝑥 = 2𝑥 + 3 − (𝑥 2 −1) = −𝑥 2 + 2𝑥 + 4 , and the domain is (−∞, ∞). ( 𝑓 𝑔) (𝑥) = 2𝑥 + 3 (𝑥 2 −1) = 2𝑥 3 − 3𝑥 2 − 2𝑥 − 3, and the domain is (−∞, ∞). 2𝑥+ 3 (𝑔/𝑓) (𝑥) = , and the domain is 𝑅 − {−1,1}. (𝑥 2 −1) 5 Example 2: 𝑓 𝑥 = 𝑥 2 − 9 , 𝑔 𝑥 =𝑥+ 3 Find and simplify expressions for the following functions. In addition, find the domain of each of these functions. 1. 𝑓 + 𝑔 𝑥 2. 𝑓 − 𝑔 𝑥 3. 𝑓𝑔 𝑥 4. 𝑓/𝑔 𝑥 Solution 6 Example 2: 𝑓 𝑥 = 𝑥 2 − 9 , 𝑔 𝑥 =𝑥+ 3 Solution 𝑓 + 𝑔 𝑥 = 𝑥2 − 9 + 𝑥 + 3 = 𝑥2 + 𝑥 − 6 , and the domain is (−∞, ∞). 𝑓 − 𝑔 𝑥 = 𝑥 2 − 9 − 𝑥 + 3 = 𝑥 2 − 𝑥 − 12 , and the domain is (−∞, ∞). 𝑓 𝑔 𝑥 = 𝑥 2 − 9 𝑥 + 3 = 𝑥 3 + 3𝑥 2 − 9𝑥 − 27 , and the domain is (−∞, ∞). 𝑥 2 −9 (𝑥 −3)(𝑥+3) (𝑔/𝑓) (𝑥) = = = 𝑥 − 3 , and the domain is 𝑅 − {−3} 𝑥+ 3 𝑥+ 3 7 Example 3: 𝑓 𝑥 = 𝑥 2 − 3 , 𝑔 𝑥 = 2𝑥 2 + 3𝑥 Find and simplify expressions for the following functions. In addition, find the domain of each of these functions. 1. 𝑓 + 𝑔 𝑥 2. 𝑓 − 𝑔 𝑥 3. 𝑓𝑔 𝑥 4. 𝑓/𝑔 𝑥 Solution 8 Example 3: 𝑓 𝑥 = 𝑥 2 − 3 , 𝑔 𝑥 = 2𝑥 2 + 3𝑥 Solution 𝑓 + 𝑔 𝑥 = 𝑥 2 − 3 + 2𝑥 2 + 3𝑥 = 3𝑥 2 + 3𝑥 − 3 , and the domain is (−∞, ∞). 𝑓 − 𝑔 𝑥 = 𝑥 2 − 3 − 2𝑥 2 + 3𝑥 = −𝑥 2 − 3𝑥 − 3 , and the domain is (−∞, ∞). 𝑓 𝑔 𝑥 = 𝑥 2 − 3 2𝑥 2 + 3𝑥 = 2𝑥 4 + 3𝑥 3 − 6𝑥 2 − 9𝑥 , and the domain is (−∞, ∞). 𝑥 2 −3 (𝑔/𝑓) (𝑥) = , and the domain is 𝑅 − {0, −3 } 2𝑥 2 + 3𝑥 2 9 Example 4: 𝑓 𝑥 = 2𝑥 2 − 4 , 𝑔 𝑥 = 𝑥 2 + 4𝑥 − 2 Find and simplify expressions for the following functions. In addition, find the domain of each of these functions. 1. 𝑓 + 𝑔 𝑥 2. 𝑓 − 𝑔 𝑥 3. 𝑓𝑔 𝑥 4. 𝑓/𝑔 𝑥 Solution 10  Example 4: 𝑓 𝑥 = 2𝑥 2 − 4 , 𝑔 𝑥 = 𝑥 2 + 4𝑥 − 2 Solution 𝑓 + 𝑔 𝑥 = 2𝑥 2 − 4 + 𝑥 2 + 4𝑥 − 2 = 3𝑥 2 + 4𝑥 − 6, and the domain is (−∞, ∞). 𝑓 − 𝑔 𝑥 = 2𝑥 2 − 4 − 𝑥 2 + 4𝑥 − 2 = 𝑥 2 − 4𝑥 − 2 , and the domain is (−∞, ∞). 𝑓 𝑔 𝑥 = 2𝑥 2 − 4 𝑥 2 + 4𝑥 − 2 = 2𝑥 4 + 8𝑥 3 − 8𝑥 2 − 16𝑥 + 8, and the domain is (−∞, ∞). 2𝑥 2 −4 (𝑔/𝑓) (𝑥) = , and the domain is R − {−2 + 6 , −2 − 6 } 𝑥 2 + 4𝑥−2 11  Composite Functions 12 Example 1: f (x) = 2x + 3, g (x) = 4x − 1 Find (𝑓 ∘ 𝑔), (𝑔 ∘ 𝑓 ), (𝑓 ∘ 𝑓), (𝑔 ∘ 𝑔) for the following functions. In addition, find the domain of each of these functions. 13 Example 1: f (x) = 2x + 3, g (x) = 4x − 1 Solution ( 𝑓 ∘ 𝑔) (𝑥) = 𝑓 (4𝑥 − 1) = 2 (4𝑥 − 1) + 3 = 8𝑥 + 1, and the domain is (−∞, ∞). (𝑔 ∘ 𝑓) (𝑥) = 𝑔 (2𝑥 + 3) = 4 (2𝑥 + 3) − 1 = 8𝑥 + 11, and the domain is (−∞, ∞). ( 𝑓 ∘ 𝑓) (𝑥) = 𝑓 (2𝑥 + 3) = 2 (2𝑥 + 3) + 3 = 4𝑥 + 9, and the domain is (−∞, ∞). (𝑔 ∘ 𝑔) (𝑥) = 𝑔 (4𝑥 − 1) = 4 (4𝑥 − 1) − 1 = 16𝑥 − 5, and the domain is −∞, ∞. 14  1 Example 2: f (x) = has domain {x | x ≠ 0} , g (x) = 2x + 4,has domain (−∞, ∞). 𝑥 Find (𝑓 ∘ 𝑔), (𝑔 ∘ 𝑓 ), (𝑓 ∘ 𝑓), (𝑔 ∘ 𝑔) for the following functions. In addition, find the domain of each of these functions. 15  1 Example 2: f (x) = has domain {x | x ≠ 0} , g (x) = 2x + 4,has domain (−∞, ∞). 𝑥 Solution 1 𝑓 ∘ 𝑔 𝑥 = 𝑓 2𝑥 + 4 = ,( 𝑓 ∘ 𝑔) (𝑥) is defined for 2𝑥 + 4 ≠ 0 ⇔ 𝑥 2𝑥+4 ≠ −2 , So the domain is {𝑥 | 𝑥 ≠ −2} = (−∞, −2) ∪ (−2, ∞). 1 2 (𝑔 ∘ 𝑓) (𝑥) = 𝑔 ( ) = + 4 𝑥 𝑥 , the domain is {𝑥 | 𝑥 ≠ 0} = (−∞, 0) ∪ (0, ∞). 16  1 Example 2: f (x) = has domain {x | x ≠ 0} , g (x) = 2x + 4,has domain (−∞, ∞). 𝑥 Solution 1 1 ( 𝑓 ∘ 𝑓) (𝑥) = 𝑓 ( ) = = 𝑥 , (𝑓 ∘ 𝑓) (𝑥) is defined whenever both 𝑓 (𝑥) and 𝑥 1Τ𝑥 𝑓 (𝑓 (𝑥)) are defined; that is, whenever {𝑥 | 𝑥 ≠ 0} = (−∞, 0) ∪ (0, ∞). (𝑔 ∘ 𝑔) (𝑥) = 𝑔 (2𝑥 + 4) = 2 (2𝑥 + 4) + 4 = 4𝑥 + 8 + 4 = 4𝑥 + 12, and the domain is (−∞, ∞). 17 Example 3: f (x) = 3x − 5, g (x) =𝑥 2 + 2 Find (𝑓 ∘ 𝑔), (𝑔 ∘ 𝑓 ), (𝑓 ∘ 𝑓), (𝑔 ∘ 𝑔) for the following functions. In addition, find the domain of each of these functions. 18 Example 3: f (x) = 3x − 5, g (x) =𝑥 2 + 2 Solution (𝑓 ∘ 𝑔) (𝑥) = 𝑓 (𝑥 2 + 2) = 3 (𝑥 2 + 2) − 5 = 3 𝑥 2 + 1, and the domain is (−∞, ∞). 𝑔 ∘ 𝑓 𝑥 = 𝑔 (3x − 5) = (3x − 5)2 +2 = 9𝑥 2 − 30x + 27 , and the domain is (−∞, ∞). (𝑓 ∘ 𝑓) (𝑥) = 𝑓 (3𝑥 − 5) = 3(3𝑥 − 5) − 5 = 9𝑥 − 20, and the domain is (−∞, ∞). 𝑔 ∘ 𝑔 𝑥 = 𝑔 𝑥 2 + 2 = (𝑥 2 + 2)2 +2 = 𝑥 4 + 4𝑥 2 + 6, and the domain is −∞, ∞. 19 Example 4: h (t) = t + 5, g (t) = 3𝑡 − 5 Find (ℎ ∘ 𝑔)(5), (𝑔 ∘ ℎ)(3), for the following functions. In addition, find the domain of each of these functions. 20 Example 4: h (t) = t + 5, g (t) = 3𝑡 − 5 Solution (ℎ ∘ 𝑔) (𝑡) = ℎ (3𝑡 − 5) = 3𝑡 − 5 + 5 = 3𝑡 , and the domain is (−∞, ∞). (ℎ ∘ 𝑔) (5) = 3(5) = 15 (𝑔 ∘ ℎ) (𝑡) = 𝑔 (𝑡 + 5) = 3(𝑡 + 5) − 5 = 3𝑡 + 10, and the domain is (−∞, ∞). (𝑔 ∘ ℎ) (3) = 3(3) + 10 = 19 21