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21:35:23 query 3.1.20 (was 3.1.6) (-2,5),(-1,3),(3,7),(4,12)}Is the given relation a function? Why or why not? If so what are its domain and range?
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RESPONSE --> yes, it is a function because there are no repeated inputs.
d={-2, -1, 3, 4} R = {5, 3, 7, 12}.................................................
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21:35:38 This relation is a function because every first element is paired with just one second element--there are no distinct ordered pairs with the same first element.
the domain is ( -2,-2,3,4) the range is ( 5,3,7,12) Another way of saying that this is a function is that every element of the domain appears only once in the relation.......!!!!!!!!...................................
RESPONSE --> ok
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21:44:40 query 3.1.34 (was 3.1.20) f(0), f(1), f(-1), f(-x), -f(x), f(x+1), f(2x), f(x+h) for 1 - 1 / (x+1)^2What are you expressions for f(0), f(1), f(-1), f(-x), -f(x), f(x+1), f(2x), f(x+h)?query 3.1.30. y = (3x-1)/(x+2)
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RESPONSE --> a) f(0) = 0 b) f(1) = 1 -1/(1+2)^2 = 1 - 1/9 c) f(-1) = 1 - 1/(-1+2)^2 = 1 - 1/1 = 1-1 = 0 d) f(-x) = 1 - 1/(-x+2)^2 = 1 - 1/(x^2 - 4x +4) 1- 1 / (-x-2)(-x - 2)
e) -f(x) = -(1 - 1/(x+2)^2) -1+ 1/(x+2)^2 f) f(x+1) = 1 - 1/((x+1+2)^2) 1 - 1/(x+3)^2 g) f(2x) = 1 - 1/(2x + 2)^2 = 1 - 1/(4x^2 + 8x + 4) h) f ( x + h) = 1 - 1/(x + h + 2 )^2.................................................
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21:50:07 STUDENT SOLUTION WITH INSTRUCTOR COMMENTS: f(x) = 1- 1/(x+2)^2
f(0) = 1- 1/ (0+2)^2 f(0) = 1-1/4 f(0) = 3/4 f(1) = 1- 1/ (3)^2 f(1) = 1- 1/9 f(1) = 8/9 f(-1) = 1- 1/(-1+2)^2 f(-1)= 1-1 f(-1)= 0 f(-x)= 1- 1/(-x+2)^2 f(-x)= 1 -1/ (x^2-4x+4) -f(x) = -(1- 1/(x+2)^2) -f(x)= -(1 - 1/ (x^2+4x+4)) -f(x) = (1/(x^2 + 4x + 4)) - 1 ** Your answer is right but you can leave it in factored form: f(-x) = -(1 - 1/(x+2)^2) = -1 + 1 / (x+2)^2. ** f(x+1) = 1- 1/((x+1) + 2)^2. This can be expanded as follows, but the expansion is not necessary at this point: = 1- 1/ ((x+1)^2 +2(x+1) + 2(x+1)+4) = 1- 1/ ((x+1)^2 +8x+8) = 1- 1/ (x^2+2x+1+8x+8) = 1- 1/(x^2 + 10x +9) ** Good algebra, and correct, but again no need to expand the square, though it is perfectly OK to do so. ** f(2x)= 1-1/(2x+2)^2 = 1- 1/(4x^2+8x+4) ** same comment ** f(x+h)= 1- 1/((x+h)+2)^2 = 1- 1/((x+h)^2 + 4(x+h) + 4) = 1- 1/ (x^2 + 2xh + h^2+4x+4h+4) ** same comment **......!!!!!!!!...................................
RESPONSE --> I see. I didn't work out f(0) because I automatically thought division by 0 wasn't defined, which isn't the case here because it is (0 + 2)^2
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21:51:45 query 3.1.44 (was 3.1.30)
Is y = (3x-1)/(x+2) the equation of a function?......!!!!!!!!...................................
RESPONSE --> yes
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21:52:48 ** This is a function. Any value of x will give you one single value of y, and all real numbers x except -2 are in the domain. So for all x in the domain this is a function. **
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RESPONSE --> I think I understand what you are saying. This doesn't come naturally to me.
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21:54:58 query 3.1.54 (was 3.1.40). G(x) = (x+4)/(x^3-4x)
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RESPONSE --> to find domain we need to factor out the denominator:
x^3 - 4x cannot = 0 x(x^2 - 4) cannot = 0 x(x+2)(x-2) cannot = 0 domain: {x|x cannot equal 0, -2 or 2}.................................................
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21:55:24 ** Starting with
g(x) = (x+4) / (x^3-4x) we factor x out of the denominator to get g(x)= (x+4) / (x (x^2-4)) then we factor x^2 - 4 to get g(x) = (x+4) / (x(x-2)(x+2)). The denominator is zero when x = 0, 2 or -2. The domain is therefore all real numbers such that x does not equal {0,2,-2}. **......!!!!!!!!...................................
RESPONSE --> right
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21:57:25 **** query 3.2.12 (was 3.1.50). Pos incr exp fnDoes the given graph depict a function? Explain how you determined whether or not the graph depicts a function.If the graph does depict a function then what are the domain and range of this function?
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RESPONSE --> yes, the graph is a function because when you do the vertical line test, each line only crosses the graph once.
domain is any real # range = y>1.................................................
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21:59:10 using the vertical line test we determine this is a function, since any given vertical line intersects the graph in exactly one point.
The function extends all the way to the right and to the left, and there are no breaks, so the domain consists of all real numbers. The range consists of all possible y values. The function takes all y values greater than zero so the range is {y | y>0}, expressed in interval notation as (0, infinity). The y intercept is (0,1); there is no x intercept but the negative x axis is an asymptote. This graph has no symmetery.......!!!!!!!!...................................
RESPONSE --> I see. I thought the range was y>1 because the graph intersected at (0,1) but I see what you are saying here.
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21:59:41 query 3.2.16 (was 3.1.54) Circle rad 2 about origin.
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RESPONSE --> not a function by the vertical line test.
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21:59:50 Using the vertical line test we see that every vertical line lying between x = -2 and x = 2, not inclusive of x = -2 and x =2, intersects the graph in two points. So this is not a function
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RESPONSE --> ok
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22:03:59 query 3.2.22 (was 3.1.60). Downward hyperbola vertex (1,2 5).Does the given graph depict a function? Explain how you determined whether or not the graph depicts a function.If the graph does depict a function then what are the domain and range of this function?
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RESPONSE --> yes, it is a function by the vertical line test.
domain: {x|-1<=x<=2} range: y<= 5.................................................
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22:05:22 Every vertical line intersects the graph at exacty one point so the graph depicts a function.
The function extends to the right and to the left without breaks so the domain consists of all real numbers. The range consists of all possible y values.......!!!!!!!!...................................
RESPONSE --> I don't understand what you mean about extending to the left and right without breaks, therefore the domain consists of all real numbers. Can you clarify?
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22:09:24 query 3.1.82 (was 3.1.70). f(x) =(2x - B) / (3x + 4).
If f(0) = 2 then what is the value of B?......!!!!!!!!...................................
RESPONSE --> f(x) = (2x - B)/(3x+4) f(2) = 1/2
(2(2) - B) / (3*2+4) (4-B)/6+4 4-B / 10 1/2 = 4 -B/10 10=8 - 2B -2B = 2 B = -1.................................................
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22:10:13 If f(0) = 2 then we have
2 = (2 * 0 - B) / (3 * 0 + 4) = -B / 4, so that B = -4 * 2 = -8. if f(2)=1/2 what is value of B? If f(2) = 1/2 then we have 1/2 = ((2*2)-B) / ((3*2)+4) 1/2 = (4-B) / 10 5 = 4-B 1=-B B=-1 **......!!!!!!!!...................................
RESPONSE --> OK
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22:18:38 query 3.1.90 (was 3.1.80). H(x) = 20 - 13 x^2 (falling rock on Jupiter)What are the heights of the rock at 1, 1.1, 1.2 seconds?When is the rock at each altitude: 15 m, 10 m, 5 m.When does the rock strike the ground?
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RESPONSE --> H(1) = 20 - 13(1)^2 = 20-13 = 7
H(1.1) = 20 - 13 (1.1)^2 = 20 - 15.73 = 4.27 H(1.2) = 20 - 13(1.2)^2 = 1.28 h(x) = 15 15 = 20-13x^2 -5 = -13x^2 x^2 = 5/13 x = .620 sec h(x) = 10 10 = 20 - 13x^2 -10 = -13x^2 10/13 = x^2 x = 0.877 sec h(x) = 5 5 = 20-13x^2 -15 = -13x^2 15/13 = x^2 x = 1.07 sec h(x) = ground 0=20-13x^2 -20 = -13x^2 20/13 = x^2 x^2 = 20/13 = 1.24 sec.................................................
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22:19:38 GOOD STUDENT SOLUTION: The height at t = 1 is
H(1) = 20-13 H(1) = 7m The height at t = 1.1 is H(1.1)= 20-13(1.1)^2 = 20-13(1.21) = 20-15.73 H(1.1)= 4.27m. The height at t = 1.2 is H(1.2)= 20 - 13*(1.2)^2 = 20- 13 *(1.44) = 20-18.72 H(1.2) = 1.28m. The rock is at altitude 15 m when H(x) = 15: 15=20-13x^2 -5=-13x^2 5/13= x^2 x= +- .62 .62sec. The rock is at altitude 10 m when H(x) = 10: 10=20-13x^2 -10=-13x^2 10/13 = x^2 x= +-.88 .88sec. The rock is at 5 meter heigh when H(x) = 5: 5=20-13x^2 -15 = -13x^2 15/13=x^2 x= +- 1.07 1.07sec. To find when the rock strikes the ground let y = 0 and we get 0= 20-13x^2. Adding -20 to both sides we have -20=-13x^2. Multiplying both sides by -1/13 we get 20/13=x^2. Taking the square root of both sides we obtain the approximate value of x: x=+-1.24 We conclude that x = 1.24sec. when the rock strikes the ground **......!!!!!!!!...................................
RESPONSE --> yes!!
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22:21:59 query 3.1.90. Sales S vs. advertising expenditures A. 335 339 337 343 341 350 351 vs. 20, 22, 22.5, 24, 24, 27, 28.3 in thousands of dollars.Does the given table describe a function? Why or why not?What two points on your straight line did you pick and what is the resulting equation?What is the meaning of the slope of this line?Give your equation as a function and give the domain of the function.What is the predicted sales if the expenditures is $25,000?
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RESPONSE --> yes, it is a function because no input (domain) is repeated.
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22:23:07 The table does not describe a function because ordered pairs that have the same first element and a different second element. Specifically 24,000 is paired with both 343,000 and 341,000.
I picked the points (20000,335000) (27000,350000). INSTRUCTOR COMMENT: These are data points, not points on the best-fit straight line graph. You should have sketched your line then picked two points on the line, and the line will almost never pass through data points. STUDENT SOLUTION CONTINUED: The slope between these points is slope = (350000-335000)/(27000-20000) = 15000/7000 = 15/7 = 2.143 approximately. Our equation, using this slope and the first chosen point, is therefore y-335000=2.143(x-20000) y- 335 = 2.143x-42857.143 y= 2.143x+29214.857 equation of the line Expressed as a function we have f(x) = 2.143x+292142.857. Predicted sales for expenditure $25000 will be f(25000) = 2.143(25000) + 292142.857 = 53575 + 292142.857 = 345717.857 We therefore have predicted sales f(25000)= $345,717.86 INSTRUCTOR COMMENT: Excellent solution, except for the fact that you used data points and not points on the best-fit line.......!!!!!!!!...................................
RESPONSE --> I see.
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