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Sagot :
To find [tex]\((f \cdot g)(x)\)[/tex] given [tex]\(f(x) = 2x - 3\)[/tex] and [tex]\(g(x) = 3x - 1\)[/tex], we need to multiply these two functions together.
Let's go through the multiplication step-by-step:
1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ f(x) = 2x - 3 \][/tex]
[tex]\[ g(x) = 3x - 1 \][/tex]
2. To find the product [tex]\((f \cdot g)(x)\)[/tex], multiply the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f \cdot g)(x) = (2x - 3)(3x - 1) \][/tex]
3. Use the distributive property (also known as FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[ (2x - 3)(3x - 1) = 2x \cdot 3x + 2x \cdot (-1) + (-3) \cdot 3x + (-3) \cdot (-1) \][/tex]
4. Perform the multiplications:
[tex]\[ 2x \cdot 3x = 6x^2 \][/tex]
[tex]\[ 2x \cdot (-1) = -2x \][/tex]
[tex]\[ (-3) \cdot 3x = -9x \][/tex]
[tex]\[ (-3) \cdot (-1) = 3 \][/tex]
5. Combine all the terms:
[tex]\[ (f \cdot g)(x) = 6x^2 - 2x - 9x + 3 \][/tex]
6. Simplify by combining like terms ([tex]\(-2x\)[/tex] and [tex]\(-9x\)[/tex]):
[tex]\[ (f \cdot g)(x) = 6x^2 - 11x + 3 \][/tex]
Thus, the product of the functions [tex]\(f(x) = 2x - 3\)[/tex] and [tex]\(g(x) = 3x - 1\)[/tex] is:
[tex]\[ (f \cdot g)(x) = 6x^2 - 11x + 3 \][/tex]
The correct choice from the given options is:
[tex]\[ 6x^2 - 11x + 3 \][/tex]
Let's go through the multiplication step-by-step:
1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ f(x) = 2x - 3 \][/tex]
[tex]\[ g(x) = 3x - 1 \][/tex]
2. To find the product [tex]\((f \cdot g)(x)\)[/tex], multiply the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f \cdot g)(x) = (2x - 3)(3x - 1) \][/tex]
3. Use the distributive property (also known as FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[ (2x - 3)(3x - 1) = 2x \cdot 3x + 2x \cdot (-1) + (-3) \cdot 3x + (-3) \cdot (-1) \][/tex]
4. Perform the multiplications:
[tex]\[ 2x \cdot 3x = 6x^2 \][/tex]
[tex]\[ 2x \cdot (-1) = -2x \][/tex]
[tex]\[ (-3) \cdot 3x = -9x \][/tex]
[tex]\[ (-3) \cdot (-1) = 3 \][/tex]
5. Combine all the terms:
[tex]\[ (f \cdot g)(x) = 6x^2 - 2x - 9x + 3 \][/tex]
6. Simplify by combining like terms ([tex]\(-2x\)[/tex] and [tex]\(-9x\)[/tex]):
[tex]\[ (f \cdot g)(x) = 6x^2 - 11x + 3 \][/tex]
Thus, the product of the functions [tex]\(f(x) = 2x - 3\)[/tex] and [tex]\(g(x) = 3x - 1\)[/tex] is:
[tex]\[ (f \cdot g)(x) = 6x^2 - 11x + 3 \][/tex]
The correct choice from the given options is:
[tex]\[ 6x^2 - 11x + 3 \][/tex]
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