course Mth 158 ?????J?????h?assignment #026
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13:49:13 3.2.66 (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)=f(x)+g(x)=|x|+x (f - g)(x)=f(x) - g(x)=|x| - x (f*g)(x)=F(x)*g(x)=|x|*x (f/g)(x)=f(x)/g(x)=|x|/x xnot=0 domain f and g is all real numbers range f is all real numbers and for g all real numbers except in the division one xnot=0 or xnot=x confidence assessment: 2
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13:49:23 ** 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 --> okay! self critique assessment: 2
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14:01:31 3.2.70 (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)=f(x)+g(x)=sqrt(x+1)+2/x (f - g)(x)=f(x) - g(x)=sqrt(x+1) - 2/x (f*g)(x)=f(x)*g(x)=sqrt(x+1)(2/x) (f/g)(x)=f(x)/g(x)=sqrt(x+1)/(2/x) domain f all real numbers >= - 1 range f all real numbers domain and range g all real numbers except 0 confidence assessment: 2
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14:01:42 ** 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 --> okay! self critique assessment: 2
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14:09:03 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))=|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)^@))=3/(9/4)+2=3/(17/4)=12/17 confidence assessment: 2
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14:17:18 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 --> okay! f(g(x))=f(sqrt(x-2)=(sqrt(x-2))^2+4=x-2+4=x+2 domain{x|x any real number} confidence assessment: 1
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14:18:40 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 --> oh..ok I get it! finish out the x+2=0 in which would be x=-2 which is x >= -2 self critique assessment: 2
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18:06:27 5.2.36 (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 --> f(g(x))=f(2/x)=(2/x)/(2/x)+3=x/(2+3x)/x=2/(2+3x) {x|xnot=0} g(f(x))=g(x/x+3)=2/(x/x+3)=2(x+3)/x or 2x+6/x {x|xnot=0} f(f(x))=f(x/x+3)=(x/x+3)/(x/+3)+3=x/(x+3) {x|xnot=-3} g(g(x))=g(2/x)=2/(2/x)=2x/2=x {x|xnot=0} confidence assessment: 2
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18:15:08 5.2.46 (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)=(x-5)+5=x g(f(x))=g(x+5)=(x+5)-5=x confidence assessment: 2
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18:15:24 ** f(g(x)) = g(x) + 5 = (x-5) + 5 = x.
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RESPONSE --> whoooah! self critique assessment: 2
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18:15:34 g(f(x)) = f(x) - 5 = (x+5) - 5 = x. **
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RESPONSE --> whooah! confidence assessment: 2
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18:21:55 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 --> H(x)=sqrt(x^2+1) f(x)=sqr(x) g(x)=x^2+1 confidence assessment: 2
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18:22:13 ** 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 --> okay! self critique assessment: 2
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18:33:24 5.2.62 (was 3.5.66). V(r) = 4/3 pi r^2 and r(t) = 2/3 t^3, t>=0. What is the requested composite function?
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RESPONSE --> V(r(t))=V(2/3t^3)=4/3pi(2/3t^3)^2 =4/3pi 4/9t^6=16/27pi t^6 confidence assessment: 2
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18:33:45 ** V(r(t)) = 4/3 pi * r(t)^2 = 4/3 pi * (2/3 t^3)^2 = 4/3 pi * (4/9 t^6) = 16/27 pi t^6. **
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RESPONSE --> whoooah! mine doesn't look that nice! self critique assessment: 2
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??????N???????g??assignment #027 027. `query 27 College Algebra 03-29-2007
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19:45:53 3.6.6. x = -20 p + 500, 0<=p<=25 What is the revenue function and what is the revenue if 20 units are sold?
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RESPONSE --> x=-20p+500; 0<=p<=25 p=500-x/20 R(x)=x(500-x/20)=-5/2x^2+25x -5/2(20)^2+25(20)=-5/2(400)+500=-1000+500=-500 confidence assessment: 1
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19:46:46 ** revenue = demand * price = x * p = (-20 p + 500) * p = -20 p^2 + 500 p If price = 24 then we get R = -20 * 24^2 + 500 * 24 = 480. **
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RESPONSE --> okay...where did I go wrong...lol??? self critique assessment: 2
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19:54:12 3.6.10. 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(x^2+y^2+2y+1) d=sqrt(x^2+(x^2-8)^2+2(x^2-8)+1) d=sqrt(x^2+x^4-16x^2+64+2x^2-16+1) d=sqrt(x^4-13x^2+49) confidence assessment: 1
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19:54:43 ** P = (x, y) is of the form (x, x^2 - 8). So the distance from P to (0, -1) is sqrt( (0 - x)^2 + (-1 - (x^2-8))^2) = sqrt(x^2 + (-7-x^2)^2) = sqrt( x^2 + 49 - 14 x^2 + x^4) = sqrt( x^4 - 13 x^2 + 49). **
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RESPONSE --> whoooooah! I got it right! self critique assessment: 2
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19:58:43 What are the values of d for x=0 and x = -1?
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RESPONSE --> d=sqrt[(0)^4-13(0)^2+49] d=sqrt(0-0+49) d=sqrt(49) d=7 d=sqrt[(-1)^4-13(-1)^2+49] d=sqrt(4-13+49) d=sqrt(40) d=2sqrt(10) confidence assessment: 1
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20:00:24 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 --> ok I got the first one right! and I am going over the second to see where I went wrong. ah...its sqrt(4+13+49)...got it self critique assessment: 2
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20:02:42 3.6. 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 --> okay??? confidence assessment: 0
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20:19:15 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 --> okay? self critique assessment: 2
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20:19:32 What is the expression for perimeter p as a function of the radius r of the circle?
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RESPONSE --> okay!? confidence assessment: 0
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20:20:19 The perimeter of the square is 4 times the length of a side which is 4 * 2r = 8r. **
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RESPONSE --> okay!? self critique assessment: 2
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20:40:27 3.6.27 (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 --> d^2=d1^2+d2^2 d^2=(2-30t)^2+(3-40t)^2 d(t)=sqrt[(2-30t)^2+(3-40t)^2] d(t)=sqrt(4-120t+9-240T+1600t^2) d(t)=sqrt(2500t^2-360t+13) confidence assessment: 2
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20:40:39 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 --> whoooah! self critique assessment: 2
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z?????G?F??y?assignment #028 028. Query 28 College Algebra 03-29-2007 "