At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine [tex]\((p \circ q)(x)\)[/tex], we need to find [tex]\(p(q(x))\)[/tex], which involves substituting [tex]\(q(x)\)[/tex] into [tex]\(p(x)\)[/tex].
Given the functions:
[tex]\[ p(x) = 2x^2 - 4x \][/tex]
[tex]\[ q(x) = x - 3 \][/tex]
First, we substitute [tex]\( q(x) \)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[ p(q(x)) = p(x-3) \][/tex]
This means we replace every [tex]\( x \)[/tex] in [tex]\( p(x) \)[/tex] with [tex]\( x-3 \)[/tex]:
[tex]\[ p(x-3) = 2(x-3)^2 - 4(x-3) \][/tex]
Now, let's break this down step-by-step:
1. Calculate [tex]\((x-3)^2\)[/tex]:
[tex]\[ (x-3)^2 = x^2 - 6x + 9 \][/tex]
2. Multiply this result by 2:
[tex]\[ 2(x-3)^2 = 2(x^2 - 6x + 9) = 2x^2 - 12x + 18 \][/tex]
3. Calculate [tex]\(-4(x-3)\)[/tex]:
[tex]\[ -4(x-3) = -4x + 12 \][/tex]
4. Combine these results:
[tex]\[ p(x-3) = 2x^2 - 12x + 18 - 4x + 12 \][/tex]
[tex]\[ p(x-3) = 2x^2 - 12x - 4x + 18 + 12 \][/tex]
[tex]\[ p(x-3) = 2x^2 - 16x + 30 \][/tex]
Therefore, [tex]\((p \circ q)(x) = 2x^2 - 16x + 30\)[/tex].
So, looking at the options provided:
[tex]\[ 2 x^2 - 4 x + 12 \][/tex]
[tex]\[ 2 x^2 - 16 x + 18 \][/tex]
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
[tex]\[ 2 x^2 - 16 x + 15 \][/tex]
The correct answer is:
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
Given the functions:
[tex]\[ p(x) = 2x^2 - 4x \][/tex]
[tex]\[ q(x) = x - 3 \][/tex]
First, we substitute [tex]\( q(x) \)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[ p(q(x)) = p(x-3) \][/tex]
This means we replace every [tex]\( x \)[/tex] in [tex]\( p(x) \)[/tex] with [tex]\( x-3 \)[/tex]:
[tex]\[ p(x-3) = 2(x-3)^2 - 4(x-3) \][/tex]
Now, let's break this down step-by-step:
1. Calculate [tex]\((x-3)^2\)[/tex]:
[tex]\[ (x-3)^2 = x^2 - 6x + 9 \][/tex]
2. Multiply this result by 2:
[tex]\[ 2(x-3)^2 = 2(x^2 - 6x + 9) = 2x^2 - 12x + 18 \][/tex]
3. Calculate [tex]\(-4(x-3)\)[/tex]:
[tex]\[ -4(x-3) = -4x + 12 \][/tex]
4. Combine these results:
[tex]\[ p(x-3) = 2x^2 - 12x + 18 - 4x + 12 \][/tex]
[tex]\[ p(x-3) = 2x^2 - 12x - 4x + 18 + 12 \][/tex]
[tex]\[ p(x-3) = 2x^2 - 16x + 30 \][/tex]
Therefore, [tex]\((p \circ q)(x) = 2x^2 - 16x + 30\)[/tex].
So, looking at the options provided:
[tex]\[ 2 x^2 - 4 x + 12 \][/tex]
[tex]\[ 2 x^2 - 16 x + 18 \][/tex]
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
[tex]\[ 2 x^2 - 16 x + 15 \][/tex]
The correct answer is:
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.