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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]
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