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Sagot :
Sure, let's go through each part of the problem step-by-step:
1. For [tex]\( f(x) = 9x - 3 \)[/tex] and [tex]\( x = -10 \)[/tex]:
- Substitute [tex]\( x = -10 \)[/tex] into the function:
[tex]\[ f(-10) = 9(-10) - 3 = -90 - 3 = -93 \][/tex]
2. For [tex]\( f(x) = -2x^3 - x \)[/tex] and [tex]\( x = -4 \)[/tex]:
- Substitute [tex]\( x = -4 \)[/tex] into the function:
[tex]\[ f(-4) = -2(-4)^3 - (-4) = -2(-64) + 4 = 128 + 4 = 132 \][/tex]
3. For [tex]\( q(x) = \frac{x+3}{x^2 + 7x + 12} \)[/tex], find [tex]\( q(3) \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[ q(3) = \frac{3 + 3}{3^2 + 7(3) + 12} = \frac{6}{9 + 21 + 12} = \frac{6}{42} = \frac{1}{7} \approx 0.14285714285714285 \][/tex]
4. If [tex]\( f(x) = -x - 11 \)[/tex], find [tex]\( 4f(-2) + 5f(1) \)[/tex]:
- Evaluate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = -(-2) - 11 = 2 - 11 = -9 \][/tex]
- Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = -(1) - 11 = -1 - 11 = -12 \][/tex]
- Calculate the expression:
[tex]\[ 4f(-2) + 5f(1) = 4(-9) + 5(-12) = -36 + (-60) = -96 \][/tex]
5. Find [tex]\( h(p + 3) \)[/tex] for [tex]\( h(x) = 2x^2 + 5x - 2 \)[/tex]:
- Substitute [tex]\( x = p + 3 \)[/tex] into the function:
[tex]\[ h(p + 3) = 2(p + 3)^2 + 5(p + 3) - 2 \][/tex]
- Expand [tex]\( (p + 3)^2 \)[/tex]:
[tex]\[ (p + 3)^2 = p^2 + 6p + 9 \][/tex]
- Substitute and simplify:
[tex]\[ h(p + 3) = 2(p^2 + 6p + 9) + 5(p + 3) - 2 \][/tex]
[tex]\[ = 2p^2 + 12p + 18 + 5p + 15 - 2 \][/tex]
[tex]\[ = 2p^2 + 17p + 31 \][/tex]
- Assuming [tex]\( p = 0 \)[/tex]:
[tex]\[ h(3) = 2(3)^2 + 5(3) - 2 = 2(9) + 15 - 2 = 18 + 15 - 2 = 31 \][/tex]
6. Find [tex]\( f(2) \)[/tex] where [tex]\( f(x) = |x - 3.7| + 2 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(2) = |2 - 3.7| + 2 = | -1.7 | + 2 = 1.7 + 2 = 3.7 \][/tex]
7. Given [tex]\( f(x) = |5.8| + x \)[/tex], solve for [tex]\( f(3) \)[/tex]:
- Note that the absolute value is a constant [tex]\( |5.8| = 5.8 \)[/tex]:
[tex]\[ f(3) = 5.8 + 3 = 8.8 \][/tex]
In summary, the results are:
1. [tex]\( f(-10) = -93 \)[/tex]
2. [tex]\( f(-4) = 132 \)[/tex]
3. [tex]\( q(3) = 0.14285714285714285 \)[/tex]
4. [tex]\( 4f(-2) + 5f(1) = -96 \)[/tex]
5. [tex]\( h(p + 3) = 31 \)[/tex]
6. [tex]\( f(2) = 3.7 \)[/tex]
7. [tex]\( f(3) = 8.8 \)[/tex]
These are the detailed steps to the solutions.
1. For [tex]\( f(x) = 9x - 3 \)[/tex] and [tex]\( x = -10 \)[/tex]:
- Substitute [tex]\( x = -10 \)[/tex] into the function:
[tex]\[ f(-10) = 9(-10) - 3 = -90 - 3 = -93 \][/tex]
2. For [tex]\( f(x) = -2x^3 - x \)[/tex] and [tex]\( x = -4 \)[/tex]:
- Substitute [tex]\( x = -4 \)[/tex] into the function:
[tex]\[ f(-4) = -2(-4)^3 - (-4) = -2(-64) + 4 = 128 + 4 = 132 \][/tex]
3. For [tex]\( q(x) = \frac{x+3}{x^2 + 7x + 12} \)[/tex], find [tex]\( q(3) \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[ q(3) = \frac{3 + 3}{3^2 + 7(3) + 12} = \frac{6}{9 + 21 + 12} = \frac{6}{42} = \frac{1}{7} \approx 0.14285714285714285 \][/tex]
4. If [tex]\( f(x) = -x - 11 \)[/tex], find [tex]\( 4f(-2) + 5f(1) \)[/tex]:
- Evaluate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = -(-2) - 11 = 2 - 11 = -9 \][/tex]
- Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = -(1) - 11 = -1 - 11 = -12 \][/tex]
- Calculate the expression:
[tex]\[ 4f(-2) + 5f(1) = 4(-9) + 5(-12) = -36 + (-60) = -96 \][/tex]
5. Find [tex]\( h(p + 3) \)[/tex] for [tex]\( h(x) = 2x^2 + 5x - 2 \)[/tex]:
- Substitute [tex]\( x = p + 3 \)[/tex] into the function:
[tex]\[ h(p + 3) = 2(p + 3)^2 + 5(p + 3) - 2 \][/tex]
- Expand [tex]\( (p + 3)^2 \)[/tex]:
[tex]\[ (p + 3)^2 = p^2 + 6p + 9 \][/tex]
- Substitute and simplify:
[tex]\[ h(p + 3) = 2(p^2 + 6p + 9) + 5(p + 3) - 2 \][/tex]
[tex]\[ = 2p^2 + 12p + 18 + 5p + 15 - 2 \][/tex]
[tex]\[ = 2p^2 + 17p + 31 \][/tex]
- Assuming [tex]\( p = 0 \)[/tex]:
[tex]\[ h(3) = 2(3)^2 + 5(3) - 2 = 2(9) + 15 - 2 = 18 + 15 - 2 = 31 \][/tex]
6. Find [tex]\( f(2) \)[/tex] where [tex]\( f(x) = |x - 3.7| + 2 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(2) = |2 - 3.7| + 2 = | -1.7 | + 2 = 1.7 + 2 = 3.7 \][/tex]
7. Given [tex]\( f(x) = |5.8| + x \)[/tex], solve for [tex]\( f(3) \)[/tex]:
- Note that the absolute value is a constant [tex]\( |5.8| = 5.8 \)[/tex]:
[tex]\[ f(3) = 5.8 + 3 = 8.8 \][/tex]
In summary, the results are:
1. [tex]\( f(-10) = -93 \)[/tex]
2. [tex]\( f(-4) = 132 \)[/tex]
3. [tex]\( q(3) = 0.14285714285714285 \)[/tex]
4. [tex]\( 4f(-2) + 5f(1) = -96 \)[/tex]
5. [tex]\( h(p + 3) = 31 \)[/tex]
6. [tex]\( f(2) = 3.7 \)[/tex]
7. [tex]\( f(3) = 8.8 \)[/tex]
These are the detailed steps to the solutions.
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