Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Ask your questions and receive precise answers from experienced professionals across different disciplines. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Certainly! Let's solve the problem step by step.
Given the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
[tex]\[ f(x) = x^2 + 2x - 3 \][/tex]
[tex]\[ g(x) = 3x - 4 \][/tex]
We need to find the composition [tex]\( (f \circ g)(x) \)[/tex], which is essentially [tex]\( f(g(x)) \)[/tex]. This means that we will first apply [tex]\( g(x) \)[/tex] and then apply [tex]\( f \)[/tex] to the result of [tex]\( g(x) \)[/tex].
1. Calculate [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 3x - 4 \][/tex]
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(3x - 4) \][/tex]
3. Replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( 3x - 4 \)[/tex]:
[tex]\[ f(3x - 4) = (3x - 4)^2 + 2(3x - 4) - 3 \][/tex]
4. Expand [tex]\( (3x - 4)^2 \)[/tex]:
[tex]\[ (3x - 4)^2 = 9x^2 - 24x + 16 \][/tex]
5. Calculate [tex]\( 2(3x - 4) \)[/tex]:
[tex]\[ 2(3x - 4) = 6x - 8 \][/tex]
6. Combine all the terms:
[tex]\[ f(3x - 4) = 9x^2 - 24x + 16 + 6x - 8 - 3 \][/tex]
7. Simplify the expression:
[tex]\[ f(3x - 4) = 9x^2 - 18x + 5 \][/tex]
Thus, the composite function [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[ (f \circ g)(x) = 9x^2 - 18x + 5 \][/tex]
To verify, let's evaluate [tex]\( (f \circ g)(x) \)[/tex] for a range of integer values of [tex]\( x \)[/tex] from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex]:
[tex]\[ \begin{align*} (f \circ g)(-10) & = 9(-10)^2 - 18(-10) + 5 = 900 + 180 + 5 = 1085, \\ (f \circ g)(-9) & = 9(-9)^2 - 18(-9) + 5 = 729 + 162 + 5 = 896, \\ (f \circ g)(-8) & = 9(-8)^2 - 18(-8) + 5 = 576 + 144 + 5 = 725, \\ (f \circ g)(-7) & = 9(-7)^2 - 18(-7) + 5 = 441 + 126 + 5 = 572, \\ (f \circ g)(-6) & = 9(-6)^2 - 18(-6) + 5 = 324 + 108 + 5 = 437, \\ (f \circ g)(-5) & = 9(-5)^2 - 18(-5) + 5 = 225 + 90 + 5 = 320, \\ (f \circ g)(-4) & = 9(-4)^2 - 18(-4) + 5 = 144 + 72 + 5 = 221, \\ (f \circ g)(-3) & = 9(-3)^2 - 18(-3) + 5 = 81 + 54 + 5 = 140, \\ (f \circ g)(-2) & = 9(-2)^2 - 18(-2) + 5 = 36 + 36 + 5 = 77, \\ (f \circ g)(-1) & = 9(-1)^2 - 18(-1) + 5 = 9 + 18 + 5 = 32, \\ (f \circ g)(0) & = 9(0)^2 - 18(0) + 5 = 0 + 0 + 5 = 5, \\ (f \circ g)(1) & = 9(1)^2 - 18(1) + 5 = 9 - 18 + 5 = -4, \\ (f \circ g)(2) & = 9(2)^2 - 18(2) + 5 = 36 - 36 + 5 = 5, \\ (f \circ g)(3) & = 9(3)^2 - 18(3) + 5 = 81 - 54 + 5 = 32, \\ (f \circ g)(4) & = 9(4)^2 - 18(4) + 5 = 144 - 72 + 5 = 77, \\ (f \circ g)(5) & = 9(5)^2 - 18(5) + 5 = 225 - 90 + 5 = 140, \\ (f \circ g)(6) & = 9(6)^2 - 18(6) + 5 = 324 - 108 + 5 = 221, \\ (f \circ g)(7) & = 9(7)^2 - 18(7) + 5 = 441 - 126 + 5 = 320, \\ (f \circ g)(8) & = 9(8)^2 - 18(8) + 5 = 576 - 144 + 5 = 437, \\ (f \circ g)(9) & = 9(9)^2 - 18(9) + 5 = 729 - 162 + 5 = 572, \\ (f \circ g)(10) & = 9(10)^2 - 18(10) + 5 = 900 - 180 + 5 = 725. \end{align*} \][/tex]
Therefore, the values of [tex]\((f \circ g)(x)\)[/tex] for [tex]\(x\)[/tex] in the range [tex]\([-10, 10]\)[/tex] are:
[tex]\[ [1085, 896, 725, 572, 437, 320, 221, 140, 77, 32, 5, -4, 5, 32, 77, 140, 221, 320, 437, 572, 725] \][/tex]
Given the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
[tex]\[ f(x) = x^2 + 2x - 3 \][/tex]
[tex]\[ g(x) = 3x - 4 \][/tex]
We need to find the composition [tex]\( (f \circ g)(x) \)[/tex], which is essentially [tex]\( f(g(x)) \)[/tex]. This means that we will first apply [tex]\( g(x) \)[/tex] and then apply [tex]\( f \)[/tex] to the result of [tex]\( g(x) \)[/tex].
1. Calculate [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 3x - 4 \][/tex]
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(3x - 4) \][/tex]
3. Replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( 3x - 4 \)[/tex]:
[tex]\[ f(3x - 4) = (3x - 4)^2 + 2(3x - 4) - 3 \][/tex]
4. Expand [tex]\( (3x - 4)^2 \)[/tex]:
[tex]\[ (3x - 4)^2 = 9x^2 - 24x + 16 \][/tex]
5. Calculate [tex]\( 2(3x - 4) \)[/tex]:
[tex]\[ 2(3x - 4) = 6x - 8 \][/tex]
6. Combine all the terms:
[tex]\[ f(3x - 4) = 9x^2 - 24x + 16 + 6x - 8 - 3 \][/tex]
7. Simplify the expression:
[tex]\[ f(3x - 4) = 9x^2 - 18x + 5 \][/tex]
Thus, the composite function [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[ (f \circ g)(x) = 9x^2 - 18x + 5 \][/tex]
To verify, let's evaluate [tex]\( (f \circ g)(x) \)[/tex] for a range of integer values of [tex]\( x \)[/tex] from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex]:
[tex]\[ \begin{align*} (f \circ g)(-10) & = 9(-10)^2 - 18(-10) + 5 = 900 + 180 + 5 = 1085, \\ (f \circ g)(-9) & = 9(-9)^2 - 18(-9) + 5 = 729 + 162 + 5 = 896, \\ (f \circ g)(-8) & = 9(-8)^2 - 18(-8) + 5 = 576 + 144 + 5 = 725, \\ (f \circ g)(-7) & = 9(-7)^2 - 18(-7) + 5 = 441 + 126 + 5 = 572, \\ (f \circ g)(-6) & = 9(-6)^2 - 18(-6) + 5 = 324 + 108 + 5 = 437, \\ (f \circ g)(-5) & = 9(-5)^2 - 18(-5) + 5 = 225 + 90 + 5 = 320, \\ (f \circ g)(-4) & = 9(-4)^2 - 18(-4) + 5 = 144 + 72 + 5 = 221, \\ (f \circ g)(-3) & = 9(-3)^2 - 18(-3) + 5 = 81 + 54 + 5 = 140, \\ (f \circ g)(-2) & = 9(-2)^2 - 18(-2) + 5 = 36 + 36 + 5 = 77, \\ (f \circ g)(-1) & = 9(-1)^2 - 18(-1) + 5 = 9 + 18 + 5 = 32, \\ (f \circ g)(0) & = 9(0)^2 - 18(0) + 5 = 0 + 0 + 5 = 5, \\ (f \circ g)(1) & = 9(1)^2 - 18(1) + 5 = 9 - 18 + 5 = -4, \\ (f \circ g)(2) & = 9(2)^2 - 18(2) + 5 = 36 - 36 + 5 = 5, \\ (f \circ g)(3) & = 9(3)^2 - 18(3) + 5 = 81 - 54 + 5 = 32, \\ (f \circ g)(4) & = 9(4)^2 - 18(4) + 5 = 144 - 72 + 5 = 77, \\ (f \circ g)(5) & = 9(5)^2 - 18(5) + 5 = 225 - 90 + 5 = 140, \\ (f \circ g)(6) & = 9(6)^2 - 18(6) + 5 = 324 - 108 + 5 = 221, \\ (f \circ g)(7) & = 9(7)^2 - 18(7) + 5 = 441 - 126 + 5 = 320, \\ (f \circ g)(8) & = 9(8)^2 - 18(8) + 5 = 576 - 144 + 5 = 437, \\ (f \circ g)(9) & = 9(9)^2 - 18(9) + 5 = 729 - 162 + 5 = 572, \\ (f \circ g)(10) & = 9(10)^2 - 18(10) + 5 = 900 - 180 + 5 = 725. \end{align*} \][/tex]
Therefore, the values of [tex]\((f \circ g)(x)\)[/tex] for [tex]\(x\)[/tex] in the range [tex]\([-10, 10]\)[/tex] are:
[tex]\[ [1085, 896, 725, 572, 437, 320, 221, 140, 77, 32, 5, -4, 5, 32, 77, 140, 221, 320, 437, 572, 725] \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
