Help if your good at maths
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Answer:
a) f(0)=4
b) f(-2)=2
c) x∈{-8,-4,4,8}
d) x∈{-9,-3,3,9}
e) x∈(-∞,-9)∪(-3,3)∪(9,∞).
f) x∈[-9,-3]∪[3,9]
g) Domain is x∈[-10,10]. Range is y∈[-4,4]. Zeros are at elements of x in {-9,-3,3,9}
h) Increasing on x∈ (-6,0) ∪ (6,10) and
decreasing on x∈ (-10,-6) ∪ (0,6).
Step-by-step explanation:
a) f(0) means what is the y-coordinate of the point at x=0.
So find 0 on the x-axis. I'm going to go up because the curve is at y=4 there.
Conclusion: f(0)=4.
b) f(-2) means what is the y-coordinate of the point at x=-2.
So find -2 on the x-axis. I'm going to go up because the curve is at y=2 there.
Conclusion: f(-2)=2.
c) f(x)=-2 means what is x when y=-2. So go on the y-axis and find -2. Now anything on that line y=-2 (horizontal line going through y=-2 on the y-axis) we need to look at.
There are 4 value we need to look at then.
x=-8
x=-4
x=4
x=8
At all of these the y-coordinate is -2.
d) f(x)=0 means what is x when y=0. So go on the y-axis and find 0. Now anything on that line y=0 (horizontal line going through y=0 on the y-axis; also knowing as the x-axis for y=0) we need to look at.
There are 4 values we need to look at then.
x=-9
x=-3
x=3
x=9
e) f(x)>0 means where is the curve above the x-axis.
The curve is above the x-axis:
- Before x=-9
- Between x=-3 and x=3
- After x=9
The interval notation is:
(-∞,-9)∪(-3,3)∪(9,∞).
d) f(x)≥0 means where is the curve below or on the x-axis (also known as y=0).
The curve is below or on the x-axis:
- Between -9 and -3 (inclusive of both endpoints because they include y=0).
- Between 3 and 9 (inclusive of both endpoints because they include y=0).
The interval notation is:
[-9,-3]∪[3,9]
g) The domain is where the curve exists for the x-values.
The domain is all real numbers between -10 and 10 (inclusive of both).
The curve starts at x=-10 and stops at x=10. The function exists a y value for any number between -10 and 10 (including both).
Interval notation is:
[-10,10]
The range is where the curve exists for the y-values.
Looking from bottom to top I see that it starts at y=-4 and stops at y=4. I notice the curve exists at some point between those two horizontal lines. It also exists at both of those endpoints" -4 and 4.
The range is between -4 and 4 (including both).
Interval notation is:
[-4,4]
The zeros are where the graph crosses the x-axis.
The graph crosses the x-axis at:
x=-9
x=-3
x=3
x=9
h) Reading left to right the graph increases when you see the curve going up.
I see this from x=-6 to x=0 (exclusive of both).
I see this from x=6 to x=10 (exclusive of both).
So interval notation is (-6,0) ∪ (6,10).
Reading left to right the graph decreases when you see the curve going down.
I see this from x=-10 to x=-6 ( exclusive of both).
I see this from x=0 to x=6 (exclusive of both).
1.
Assume the people A, B, C, D and E are sitting in a row.
since there are 5 positions, they can sit in 5*4*3*2*1=120 ways.
consider 1 certain sitting position, for example ABCDE, that is B has A to his right, C to his left. D has C to his right and E to his left. And so on.
the positions
ABCDE
BCDEA
CDEAB
DEABC
EABCD
while in a row are different positions, in a round table they are the same thing.
(read the letters starting from A, and when the letters finish, continue reading from the beginning of the row. They all read "ABCDE")
This means that any arrangement in a round table, is converted to 5 arrangements in a row.
So there are in total
arrangements of 5 people around a table.
2. check picture.
Consider the case where A is at (2, 2) and B is at (5, 6)
A can move to B through 3 horizontal units to the right, we call them E (for East) and 4 vertical units up, which we call N (for north).
the total path is 7 units.
it can be done in several ways, for example:
ENNNENE (the red path shown in the figure)
NNEEENN (the black path shown in the figure)
In total there are
paths.
Remark: 7! is the number of arrangements if the letters were different. We divide by 3! because of the 3 E's and 4! because of the 4 N's.
Another possibility could be A'(3, -5), B'(8,-3).
From A' to B' we go by a total of 5 E and 2 N, so there are
paths in total.
there is one more possibility
C.
a) consider the letters {V,E,C,T,O,R}
we can form 6*5*4*3*2*1=720 words, as the first digit can be any of the 6 letters, combined with 5 for the second letter and so on.
b) the difference with a is that we have 5 letters and we have 2 T.
we have in total 5*4*3*2*1=120 arrangements of these letters, if the 2 T's were considered as separate.
consider the arrangements:
and 
.
We read bth as just TRUST, but the permutation formula 5*4*3*2*1 considers these as 2 different.
this is why we need to divide 120 by 2, to get the actual number of words that can be formed. So the number is 60.
Answers:
1) 24
2) 7, 35 or 42. it depends on the positions of A and B, check the solution
3) a-720, b-160
Assume the people A, B, C, D and E are sitting in a row.
since there are 5 positions, they can sit in 5*4*3*2*1=120 ways.
consider 1 certain sitting position, for example ABCDE, that is B has A to his right, C to his left. D has C to his right and E to his left. And so on.
the positions
ABCDE
BCDEA
CDEAB
DEABC
EABCD
while in a row are different positions, in a round table they are the same thing.
(read the letters starting from A, and when the letters finish, continue reading from the beginning of the row. They all read "ABCDE")
This means that any arrangement in a round table, is converted to 5 arrangements in a row.
So there are in total
2. check picture.
Consider the case where A is at (2, 2) and B is at (5, 6)
A can move to B through 3 horizontal units to the right, we call them E (for East) and 4 vertical units up, which we call N (for north).
the total path is 7 units.
it can be done in several ways, for example:
ENNNENE (the red path shown in the figure)
NNEEENN (the black path shown in the figure)
In total there are
Remark: 7! is the number of arrangements if the letters were different. We divide by 3! because of the 3 E's and 4! because of the 4 N's.
Another possibility could be A'(3, -5), B'(8,-3).
From A' to B' we go by a total of 5 E and 2 N, so there are
there is one more possibility
C.
a) consider the letters {V,E,C,T,O,R}
we can form 6*5*4*3*2*1=720 words, as the first digit can be any of the 6 letters, combined with 5 for the second letter and so on.
b) the difference with a is that we have 5 letters and we have 2 T.
we have in total 5*4*3*2*1=120 arrangements of these letters, if the 2 T's were considered as separate.
consider the arrangements:
We read bth as just TRUST, but the permutation formula 5*4*3*2*1 considers these as 2 different.
this is why we need to divide 120 by 2, to get the actual number of words that can be formed. So the number is 60.
Answers:
1) 24
2) 7, 35 or 42. it depends on the positions of A and B, check the solution
3) a-720, b-160