Questions and Answers Sheet 8 Function Question #1 Given that 𝑔(𝑥) = 4𝑥 + 6, what is the value of x that makes 𝑔(𝑥) = 14 ? Answer: So for given equation: 𝑔(𝑥) = 4𝑥 + 6; 𝑔(𝑥) = 14: .14 = 4𝑥 + 6 or 4𝑥 = 14 − 6 or 4𝑥 = 8 𝑜𝑟 𝑥 = 2 The value of x is 2 Question #2 Let 𝑓(𝑥) = 4𝑥 + 5 and 𝑔(𝑥) = 4𝑥 2 − 5 . What is (𝑓 ∗ 𝑔)(𝑥 − 1) ? Answer: So, solution is: "evaluate "(𝑓 ∗ 𝑔)(𝑥) = 𝑓(𝑥) ∗ 𝑔(𝑥) (4𝑥 + 5)(4𝑥 2 − 5) = 16𝑥 3 + 20𝑥 2 − 20𝑥 − 25 "to evaluate "(𝑓 ∗ 𝑔)(𝑥 − 1) "substitute "𝑥 = 𝑥 − 1" into "(𝑓 ∗ 𝑔)(𝑥) = 16(𝑥 − 1)3 + 20(𝑥 − 1)2 − 20(𝑥 − 1) − 25 = 16(𝑥 3 − 3𝑥 2 + 3𝑥 − 1) + 20(𝑥 2 − 2𝑥 + 1) − 20𝑥 + 20 − 25 = 16𝑥 3 − 48𝑥 2 + 48𝑥 − 16 + 20𝑥 2 − 40𝑥 + 20 − 20𝑥 − 5 = 16𝑥 3 − 28𝑥 2 − 12𝑥 − 1 Question #3 What is the domain of the function 𝑓(𝑥) = 2𝑥 + 5? Answer: The definition of this function covers all x values. In other words, there are no x values that lead to the undefined behavior of this function. So the domain is (-infinity, +infinity). Question #4 If ℎ(𝑥) = −2𝑥 − 10 , what is ℎ(−4) ? Answer: We have: ℎ(𝑥) = −2𝑥 − 10 We need replace x with −4 → ℎ(−4) = −2 ∗ (−4) − 10 = +8 − 10 = −2 Question #5 What is the remainder when (2𝑥 3 + 4𝑥 2 − 32𝑥 + 18) ÷ (𝑥 + 3) ? Answer: Let 𝑓(𝑥) = 2𝑥 3 + 4𝑥 2 − 32𝑥 + 18 When f(x) is divided by (𝑥 − 𝑎), the result is f(a). So when divide by (𝑥 + 3) = (𝑥 − (−3)) the remainder is: 𝑓(−3) = 2(−27) + 4(9) − 32(−3) + 18 = −54 + 36 + 96 + 18 = 96 Question #6 In general, what is the y intercept of the function 𝐹(𝑥) = 𝑎 ∗ 𝑏 𝑥 ? Answer: The y intercept is when 𝑥 = 0 thus 𝐹(0) = 𝑎 ⋅ 𝑏 0 = 𝑎 Question #7 Is 𝑥 + 10 a factor of the function 𝑓(𝑥) = 𝑥 3 − 75𝑥 + 250 ? Answer: If 𝑥 + 10 is a factor, we can determine it using the factor theorem. The factor theorem states that 𝑥 − 𝑎 is a factor of f(x) iff and only if 𝑓(𝑎) = 0 then for 𝑥 + 10 to be a factor 𝑓(−10) = 0 𝑓(𝑥) = 𝑥 3 − 75𝑥 + 250 𝑓(−10) = (−10)3 − 75𝑥𝑥 − 10 + 250 𝑓(−10) = −1000 + 750 + 250 𝑓(−10) = 0 ∴ 𝑥 + 10" " is a factor of 𝑓(𝑥) = 𝑥 3 − 75𝑥 + 250