Assingment 26

You're doing pretty well here.

If anything is not clear let me know, and include as many specifics as possible.

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20:04:00 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 = 2x f - g = |x| - x = 0 f*g = |x| * x = x^2 f/g = |x|/x = 1x domain and range of all are all real numbers

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20:11:28 ** 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 --> You said that the abs value of x when x<0 is -x. I thought the absolute value of any number was a positive number because absolute value is the distance from the number to zero on the number line.

That is correct. If x < 0 then -x is > 0. For example if x = -3, then -x = - ( -3 ) = 3, which is the absolute value of 3.

Exactly. -4 is < 0, so if x = -4 we have | x | = - x = - (-4) = 4.

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20:15:52 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 = sqrt(x+1) + 2/x domain: x cannot = 0,-1

x can be -1 because sqrt(0) is fine as long as it isn't the denominator.

However x cannot be negative because you the sqrt of a negative is not a real number.

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20:19:28 ** 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 --> I am going to have to review this again before I can explain in depth what I did wrong.

You had the right idea but you had the rule wrong for the square root. The domain of sqrt(x) includes 0 but does not include any negative numbers.

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20:25:08 query 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 --> a: f(g(4)) first find g(4) and then use that value to find f(g) 3/4^2 +2 = 3/18=1/6 f(1/6) = |1/6 -2| = |-1.83| = 1.83 b: g(f(2)) = |2-2| = 0 g(0) = 3/0^2 + 2 = 3/2 c: f(f(1)) = |1-2| = |-1| = 1 f(1) = |1-2| = |-1| = 1 d: g(g(0) = 3/0^2 + 2 = 3/2 g(3/2) = 3/(3/2)^2 + 2 = 3/9/4 + 2 = 3/4.25 OR 3/(17/4)

3/(17/4) is a complex fraction and needs to be simplified. It gives you 3 * 4/17 = 12 / 17, as below.

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20:31:57 ** 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 --> I see we do ours differently.

You rounded off the 11/6 to 1.83; should have been 1.8333... . Otherwise your results agree with the results given here.

You should ideally understand the process both ways, but either way is fine.

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20:35:24 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 --> the domain of x^2 + 4 is the set of all real numbers the domain of sqrt (x-2) is {x|x >= 2} so the domain of the function is all real numbers >= 2

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20:36:25 ** 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 --> x -2 >= 0 would be x >= 2

That is correct.

The domain would also be [2, infinity).

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20:36:52 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 --> right

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20:56:39 query 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 --> the domain of x/x+3 is x+3=0 x cannot = -3 the domain of 2/x is (-inf, 0) U (0,inf) f(g(x)) g(x) = 2/x f(2/x) = 2/x / [(2/x) + 3] the domain is found when g(x) = -3 2/x = -3 2=-3x x=-2/3 the domain is {x|x cannot = -2/3} g(f(x)) (2/x ) / x + 3

g(f(x)) = 2 / f(x) = 2 / (x/(x+3)) = 2(x+3) / x.

x can be 0. x isn't the denominator.

It is denominators that can't be 0.

f(f(x)) = f(x/x+3) = (x/x+3) / (x/(x+3) + 3)

this is correct but needs to be simplified

x can't be -3, as you say.

The denominator x / (x+3) + 3 can't be zero either. However that doesn't occur for x = 0, but rather for x = -9/4.

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20:59:35 ** 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 0 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 --> Honestly, this is all pretty confusing for me. I am going to have to study this more.

Take another look at this problem, at my notes, and at the given solution. Let me know if you have specific questions.

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21:01:02 query 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) = (x-5) + 5 = x g(f(x)) = g (x+5) = (x+5) - 5 = x

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21:01:16 ** f(g(x)) = g(x) + 5 = (x-5) + 5 = x.

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RESPONSE --> yes!!

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21:01:21 g(f(x)) = f(x) - 5 = (x+5) - 5 = x. **

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RESPONSE --> ok

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21:04:06 query 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 --> IF: f(x) = x^2 + 1 and g(x) = sqrt(x) then f(g(x)) =f(sqrt x) = sqrt (x^2 + 1)

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21:06:00 ** 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 --> I see. I should have had g(x) = x^2 + 1

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21:07:32 query 5.1.62 (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)) = 4/3pi r^3(2/3t^3) = 8/9 pi r^3 t ^3

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21:08:56 11-06-2005 21:08:56 ** 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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21:09:01 ** 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 --> i see

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