course Mth 158
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RESPONSE --> f + g |x| + x Domain: all real numbers Range: all positive real numbers f - g |x| - x Domain: all real numbers Range: all positive real numbers f(g) |x|(x) Domain: all real numbers Range: all real numbers f/g |x|/x Domain: {x|x not = 0} Range : all real numbers
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18:34:25 ** 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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18:38:44 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: sqrt(x + 1) is x>-1 and 2/x is x not = 0 f - g sqrt(x + 1) - 2/x Domain: sqrt(x + 1) is x>-1 and 2/x is x not = 0 f * g sqrt(x + 1) * 2/x Domain: sqrt(x + 1) is x>-1 and 2/x is x not = 0 f/g sqrt(x + 1) / 2/x Domain: sqrt(x + 1) is x>-1 and 2/x is x not = 0
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18:38:48 ** 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 --> ok
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18:44:30 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 --> f(g(4)) = f(3/(4^2 + 2) = f(3/18) = |3/18 - 2| = |3/18 - 36/18| = |11/6| g(f(2)) = g(|2 - 2|) = 3/2 f(f(1)) = f(|1 - 2|) = f(|-1|) = f (1 - 2) = |-1| = 1 g(g(0)) = g(3/2) = g(3/(3/2^2 + 2) = 3/(9/4 + 2) = 12/17
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18:44:52 ** 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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18:45:30 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 --> This problem not in this section.
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18:45:35 ** 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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18:45:39 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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18:50:51 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 --> f(g(x)) Domain of g: {x|x not = 0} Domain of f: {x|x not = -3} domain of f(g(x)): {x|x not = 0,-2/3} g(f(x)) Domain of g: {x|x not = -2/3} Domain of f: {x|x not = -3} domain of f(g(x)): {x|x not = 0,-3} f(f(x)) Domain of f: {x|x not = -3} domain of f(f(x)): {x|x not = -9/4,-3} g(g(x)) Domain of g: {x|x not = 0} domain of g(g(x)): {x|x not = 0}
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18:50:55 ** 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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18:52:21 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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18:52:26 ** f(g(x)) = g(x) + 5 = (x-5) + 5 = x.
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RESPONSE --> ok
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18:52:29 g(f(x)) = f(x) - 5 = (x+5) - 5 = x. **
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RESPONSE --> ok
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18:53:02 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 --> f(x) = sqrt(x) g(x) = x^2 + 1
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18:53:05 ** 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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18:54:17 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)) V = 2/3t^3 = 4/3pi(2/3t^3)^2 V = 16/27 pi t^6
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18:54:22 ** 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 --> ok
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