Problem Set Solutions

Let's solve each of the problems one by one.

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Given functions:

$f(x) = 5x - 8$ $g(x) = 3x + 10$ $h(x) = -5x + 3$

1. $(g+h)(-1)$

$(g+h)(x) = g(x) + h(x) = (3x + 10) + (-5x + 3) = -2x + 13$ $(g+h)(-1) = -2(-1) + 13 = 2 + 13 = 15$ Answer: $15$

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2. $(h-f)(-2)$

$(h-f)(x) = h(x) - f(x) = (-5x + 3) - (5x - 8) = -10x + 11$ $(h-f)(-2) = -10(-2) + 11 = 20 + 11 = 31$ Answer: $31$

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3. $(g-f)(5)$

$(g-f)(x) = g(x) - f(x) = (3x + 10) - (5x - 8) = -2x + 18$ $(g-f)(5) = -2(5) + 18 = -10 + 18 = 8$ Answer: $8$

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4. $(f-9)(4)$

$f(4) = 5(4) - 8 = 20 - 8 = 12$ $f(4) - 9 = 12 - 9 = 3$ Answer: $3$

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5. $(f-h)(-3)$

$(f-h)(x) = f(x) - h(x) = (5x - 8) - (-5x + 3) = 10x - 11$ $(f-h)(-3) = 10(-3) - 11 = -30 - 11 = -41$ Answer: $-41$

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6. $(f+hg)(2)$

First, calculate $hg(x)$: $hg(x) = h(g(x)) = h(3x + 10) = -5(3x + 10) + 3 = -15x - 50 + 3 = -15x - 47$ Then, $(f+hg)(x) = f(x) + hg(x) = (5x - 8) + (-15x - 47) = -10x - 55$ $(f+hg)(2) = -10(2) - 55 = -20 - 55 = -75$ Answer: $-75$

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Given Functions:

$f(x) = 2x - 1$ $g(x) = 3x + 2x + 4 = 5x + 4$

7. $(f-g)(x)$

$(f-g)(x) = f(x) - g(x) = (2x - 1) - (5x + 4) = -3x - 5$ Answer: $-3x - 5$

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8. $(f-g)(0)$

$(f-g)(0) = -3(0) - 5 = -5$ Answer: $-5$

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9. $(f-9)(2)$

$f(2) = 2(2) - 1 = 4 - 1 = 3$ $f(2) - 9 = 3 - 9 = -6$ Answer: $-6$

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10. $(f-g)(-2)$

$(f-g)(-2) = -3(-2) - 5 = 6 - 5 = 1$ Answer: $1$

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Given Functions:

$f(x) = x + 1$ $g(x) = x^2 + 3x + 2$

11. $\left(\frac{f}{g}\right)(x)$

$\left(\frac{f}{g}\right)(x) = \frac{x + 1}{x^2 + 3x + 2}$ Answer: $\frac{x+1}{x^2 + 3x + 2}$

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12. $\left(\frac{f}{9}\right)(-3)$

$\frac{f(-3)}{9} = \frac{-3 + 1}{9} = \frac{-2}{9} = -\frac{2}{9}$ Answer: $-\frac{2}{9}$

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Additional Problems

1. Average Speed:

$\text{Distance} = d(x) = 6x^2 - x - 2$ $\text{Time} = t(x) = 3x - 2$ $\text{Average Speed} = \frac{d(x)}{t(x)} = \frac{6x^2 - x - 2}{3x - 2}$

Answer: $\frac{6x^2 - x - 2}{3x - 2}$

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2. Total Distance:

$\text{Speed} = s(x) = 2x - 3x - 1 = -x - 1$ $\text{Time} = t(x) = 3x - 4$ $\text{Distance} = s(x) \times t(x) = (-x - 1)(3x - 4) = -3x^2 + 4x - 3x + 4 = -3x^2 + x + 4$

Answer: $-3x^2 + x + 4$

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Would you like to dive into any of these solutions for more detail or have any questions?

Relative Questions:

1. How do you find the domain of a function after performing operations?
2. What are the implications of combining two functions via multiplication?
3. How does the operation of division of two functions differ from multiplication in terms of domain?
4. How do you find the intersection points of two functions after performing a subtraction operation?
5. How would you approach the evaluation of composite functions?
6. What is the importance of factoring in simplifying function expressions?
7. How can operations on functions be applied to real-life situations, such as in physics?
8. What are the potential challenges when dividing two polynomial functions?

Tip: When subtracting functions, always distribute the negative sign through the entire second function before combining like terms.