course Mth 158 ȈW눏Дzwғ|assignment #026
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14:31:08 query 3.1.66 (was 3.5.6). f+g, f-g, f*g and f / g for | x | and x. What are f+g, f-g, f*g and f / g and what is the domain and range of each?
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RESPONSE --> (f + g)(x) = |x| + x domain any real number (f - g)(x) = |x| - x domain any real number (f * g) (x) = |x|x domain any real number (f/g) (x) = |x|/x domain not= 0
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14:35:12 ** The domain of f is all real numbers and its range is all positive numbers. The domain of g is all real numbers and its range is all real numbers. We recall that if x < 0 it follows that | x | = -x, whereas for x > 0 we have | x | = x. The domain of f + g is all real numbers. f + g = | x | + x. Since for negative x we have | x | = -x, when x < 0 the value of f + g is zero. For x = 0 we have f + g = 0 and for x > 0 we have f + g > 0, and f + g can take any positive value. More specifically for positive x we have f + g = 2x, and for positive x 2x can take on any positive value. The range of f + g is therefore all non-negative real numbers. The domain of f - g is all real numbers. f - g = | x | - x. Since for positive x we have | x | = x, when x > 0 the value of f - g is zero. For x = 0 we have f + g = 0 and for x < 0 we have f - g > 0, and f + g can take any positive value. More specifically for negative x we have f - g = -2x, and for negative x the expression -2x can take on any positive value. The range of f - g is therefore all non-negative numbers. The domain of f * g is all real numbers. f * g = | x | * x. For x < 0 then f * g = -x * x = -x^2, which can take on any negative value. For x = 0 we have f * g = 0 and for x > 0 we have f * g = x^2, which can take on any positive value. The range of f * g is therefore all real numbers. The domain of f / g = | x | / x is all real numbers for which the denominator g is not zero. Since g = 0 when x = 0 and only for x = 0, the domain consists of all real numbers except 0. For x < 0 we have | x | / x = -x / x = -1 and for x > 0 we have | x | / x = x / x = 1. So the range of f / g consists of just the value 1 and -1; we express this as the set {-1, 1}. **
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RESPONSE --> ok
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14:47:33 query 3.1.70 (was 3.5.10). f+g, f-g, f*g and f / g for sqrt(x+1) and 2/x. What are f+g, f-g, f*g and f / g and what is the domain and range of each?
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RESPONSE --> (f + g)(x) = sqrt(x + 1) + 2/x = x + 1 + (4/x^2) = (x^3 + x^2 + 4)/x^2 domain not = 0 (f - g0 (x) = sqrt(x + 1) - (2/x) = (x^3 +x -4)/x^2 domain not= 0 (f * g)(x) = sqrt(x +1) * 2/x = (x+ 1)(4/x^2) = (x^3 + x^2)(4) = 4x^3 + 4x doamin not= 0 (f/g)(x) = (sqrt(x+ 1))/(2/x) = sqrt(x +1)(x/2) = (x + 1)(x^2/4) = 4x^3 +4x^2 domain not = 0
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14:49:53 ** The square root is always positive and the argument of the square root must be nonnegative, so sqrt(x + 1) is defined only when x+1 > 0 or x > -1. So the domain of f is all real numbers greater than or equal to -1 and its range is all positive numbers. The function g(x) = 2/x is defined for all values of x except 0, and 2/x = y means that x = 2 / y, which gives a value of x for any y except 0. So the domain of g is all real numbers except 0 and its range is all real numbers except 0. Any function obtained by combining f and g is restricted at least to a domain which works for both functions, so the domain of any combination of these functions excludes values of x which are less than -1 and x = 0. The domain will therefore be at most {-1,0) U (0, infinity). Other considerations might further restrict the domains. The domain of f + g is {-1,0) U (0, infinity). There is no further restriction on the domain. The domain of f - g is {-1,0) U (0, infinity). There is no further restriction on the domain. The domain of f * g is {-1,0) U (0, infinity). There is no further restriction on the domain. The domain of f / g = | x | / x is {-1,0) U (0, infinity) for which the denominator g is not zero. Since the denominator function g(x) = 2/x cannot be zero there is no further restriction on the domain. **
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RESPONSE --> a little too complicated.
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14:58:09 query 6.1.18 / 5.1.16 (was 3.5.20?). f(g(4)), g(f(2)), f(f(1)), g(g(0)) for |x-2| and 3/(x^2+2) Give the requested values in order and explain how you got each.
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RESPONSE --> (f(g(4)) = f(3/(4^2 + 2) f(3/18) = 1/6 |1/6 -2| = |-11/6| = 11/6 g(f(2)) = g(|2 -2|) =g(0) 3/(0+2) = 3/2 f(f(1)) = f(|1 -2|) = (|-1|) = 1 |1 -2| = |-1| = 1 g(g(0)) = g(0)=(3/(0+2) = 3/2 3/[(3/2)^2 + 2] = 3/(9/4 +2) = 3/(17/4) = 3(4/17) = 12/17
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14:58:30 ** f(g(4)) = | g(4) - 2 | = | 3 / (4^2 + 2) - 2 | = | 3/18 - 2 | = | 1/6 - 12/6 | = | -11/6 | = 11/6. g(f(2)) = 3 / (f(2)^2 + 2) = 3 / ( | 2-2 | ) ^2 + 2) 3 / (0 + 2) = 3/2. f(f(1)) = | f(1) - 2 | = | |1-2| - 2 | = | |-1 | - 2 | = | 1 - 2 | = |-1| = 1. g(g(0)) = 3 / (g(0)^2 + 2) = 3 / ( (3 / ((0^2+2)^2) ^2 + 2)) = 3 / (9/4 + 2) = 3/(17/4) = 12/17. **
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RESPONSE --> ok
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15:02:09 query 5.2.16 (was 3.5.30). Domain of f(g(x)) for x^2+4 and sqrt(x-2) What is the domain of the composite function?
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RESPONSE --> domain = any real number greater than 2
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15:02:28 ** The domain of g(x) consists of all real numbers for which x-2 >= 0, i.e., for x >= -2. The domain is expressed as {-2, infinity}.
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RESPONSE --> ok
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15:02:42 The domain of f(x) consists of all real numbers, since any real number can be squared and 4 added to the result. The domain of f(g(x)) is therefore restricted only by the requirement for g(x) and the domain is {-2, infinity}. **
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RESPONSE --> ok
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15:08:30 query 6.1.24 / 5.1.26 (was 3.5.40). f(g(x)), g(f(x)), f(f(x)), g(g(x)) for x/(x+3) and 2/x Give the four composites in the order requested and state the domain for each.
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RESPONSE --> This problem only asked for the domain. Domain:g not = 0 f not = -3 g(x) not= -2/3 domain x not = 0 and not = -2/3
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15:09:38 ** The domain of f(x) is all x except -3. The domain of g(x) is all x except 0. The domain of f(g(x)) consists of all x for which the argument of g is not zero and for which the argument of f is not -3. The argument of g is x so x cannot be zero and the argument of f is g(x) so g(x) cannot be -3. This means that 2/x = -3 is not possible. Solving this for x we find that x cannot be -2/3. The domain of f(g(x)) is therefore all real numbers except -3 and -2/3. The domain of f(f(x)) consists of all x for which the argument of the first f is not -3 and for which the argument of the second f is not -3. The argument of the second f is x so x cannot be -3 and the argument of the first f is f(x) so f(x) cannot be -3. This means that x/(x+3) = -3 is not possible. Solving this for x we find that x cannot be -9/4. The domain of f(f(x)) is therefore all real numbers except -3 and -9/4. The domain of g(f(x)) consists of all x for which the argument of f is not -3 and for which the argument of g is not 0. The argument of f is x so x cannot be -3 and the argument of g is f(x) so f(x) cannot be 0. f(x) is zero if and only if x = 0. The domain of g(f(x)) is therefore all real numbers except -3 and 0. The domain of g(g(x)) consists of all x for which the argument of the first g is not 0 and for which the argument of the second g is not 0. The argument of the second g is x so x cannot be 0 and the argument of the first g is g(x) so g(x) cannot be 0. There is no real number for which g(x) = 2/x is zero. The domain of g(g(x)) is therefore all real numbers except 0. **
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RESPONSE --> ok
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15:12:37 query 6.1.48 / 5.1.46 (was 3.5.50). f(g(x)) = g(f(x)) = x for x+5 and x-5 Show f(g(x)) = g(f(x)) = x for the given functions.
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RESPONSE --> f(g(x)) = f(x -5) g(f(x)) = g(x+5) x-5 +5 x +5 -5 =x =x
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15:12:52 ** f(g(x)) = g(x) + 5 = (x-5) + 5 = x.
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RESPONSE --> ok
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15:12:58 g(f(x)) = f(x) - 5 = (x+5) - 5 = x. **
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RESPONSE --> ok
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15:14:22 query 6.1.55 / 5.1.53 (was 3.5.60). H(x) = sqrt(x^2 + 1) = f(g(x)) Give the functions f and g such that H is the composite.
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RESPONSE --> g(x) = x^2 + 2 f(x) = sqrt (x)
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15:14:53 ** The composite f(g(x)) has 'innermost' function g(x), to which the f function is applied. The 'innermost' function of sqrt(x^2 + 1) is x^2 + 1. The square root is applied to this result. So H(x) = f(g(x)) with f(u) = sqrt(u) and g(x) = x^2 + 1. Thus f(g(x)) = sqrt(g(x)) = sqrt(x^2 + 1). **
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RESPONSE --> ok
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15:17:10 query 6.1.66 / 5.1.62 (was 3.5.66). V(r) = 4 /3 pi r^3 and r(t) = 2/3 t^3, t>=0. What is the requested composite function?
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RESPONSE --> V (r(t)) = (4/3)pi (2/3t^3)^3 = (4/3) pi ( 8/27t^9) = (32/81)pi(t^9)
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15:18:06 ** V(r(t)) = 4/3 pi * r(t)^3 = 4/3 pi * (2/3 t^3)^3 = 4/3 pi * (8/12 t^9) = 32/36 pi t^9 = 8/9 pi t^9. **
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RESPONSE --> I don't get the 8/12 part.
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lxC䚘Jظ assignment #027 027. `query 27 College Algebra 10-14-2008
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15:21:24 query 3.6.2 / 10. P = (x, y) on y = x^2 - 8. Give your expression for the distance d from P to (0, -1)
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RESPONSE --> d = sqrt[(0-x)^2 +( -1 -y)^2] = sqrt(x^2 + 1 + 2y + y^2)
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15:30:29 What are the values of d for x=0 and x = -1?
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RESPONSE --> ok x=0 d = 94 x=1 d= 82.
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15:31:48 ** If x = 0 we have sqrt( x^4 - 13 x^2 + 49) = sqrt(0^4 - 13 * 0 + 49) = sqrt(49) = 7. If x = -1 we have sqrt( x^4 - 13 x^2 + 49) = sqrt((-1)^4 - 13 * (-1) + 49) = sqrt( 64) = 8. Note that these results are the distances from the x = 0 and x = 1 points of the graph of y = x^2 - 8 to the point (0, -1). You should have a sketch of the function and you should vertify that these distances make sense. **
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RESPONSE --> I've just messed this one up all the way around.
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15:35:03 query 3.6.9 / 18 (was and remains 3.6.18). Circle inscribed in square. What is the expression for area A as a function of the radius r of the circle?
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RESPONSE --> A(x) = 2x[2(sqrt (4- x^2))} A(x) = 4x(sqrt(4 -x^2))
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15:36:43 ** A circle inscribed in a square touches the square at the midpoint of each of the square's edges; the circle is inside the square and its center coincides with the center of the square. A diameter of the circle is equal in length to the side of the square. If the circle has radius r then the square has sides of length 2 r and its area is (2r)^2 = 4 r^2. The area of the circle is pi r^2. So the area of the square which is not covered by the circle is 4 r^2 - pi r^2 = (4 - pi) r^2. **
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RESPONSE --> ok
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15:39:55 What is the expression for perimeter p as a function of the radius r of the circle?
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RESPONSE --> I don't understans the first one, so I know I didn't get this one. The answer in the book for problem 9 is different from yours, also.
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15:40:55 ** The perimeter of the square is 4 times the length of a side which is 4 * 2r = 8r. **
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RESPONSE --> This is problem 10. I was doing nine.
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15:41:27 query 3.6.19 / 27 (was 3.6.30). one car 2 miles south of intersection at 30 mph, other 3 miles east at 40 mph Give your expression for the distance d between the cars as a function of time.
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RESPONSE --> I got the answer out of the book.
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15:42:43 ** At time t the position of one car is 2 miles south, increasing at 30 mph, so its position function is 2 + 30 t. The position function of the other is 3 + 40 t. If these are the x and the y coordinates of the position then the distance between the cars is distance = sqrt(x^2 + y^2) = sqrt( (2 + 30 t)^2 + (3 + 40t)^2 ) = sqrt( 4 + 120 t + 900 t^2 + 9 + 240 t + 1600 t^2) = sqrt( 2500 t^2 + 360 t + 13). **
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RESPONSE --> The book has 2- 30t and 3- 40t. That changes the answer. Which is right?
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