Problem 1 is about relation versus function. See this link.
Question two is about slope. Steeper means more difficult. In the case of positive slope, greater slope means steeper.
Problem #1: If you have pieces of paper in one box with the numbers 0, 1, 2, 3, 4 on them and pieces of paper in a second box with the numbers 5, 6, 7, 8, 9 on them, explain how you could form five ordered pairs that would represent a function and five ordered pairs that would represent a relation. Explain your reasoning.
1. Explain how you could form five ordered pairs that would represent a function and five ordered pairs that would represent a relation.
2. Explain your reasoning.
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Problem #2 : Which of the following ski runs would present the greatest challenge? Why? Run #1, which has a vertical change of 100 feet for each horizontal change of 500 feet, or Run #2 which has a vertical change of 100 feet and a horizontal change of 1000 feet.
Find the slope of each run to justify your answer. It may be helpful to draw a picture of both runs as close to scale as possible. Explain your reasoning.
1. Which of the ski runs would present the greatest challenge?
2. Explain your reasoning.
I appreciate all help, thanks!
Well, it helps to have concrete examples and an "intuitive feel" for the concepts rather than just relying on formal definitions.
To tell the truth, you might not have a good feel for the (real) formal definitions for some time, because they are usually introduced at University level. But the concepts themselves are not hard.
Possibly the relation that's most familiar to you is the "is equal to" relation. Suppose we're only dealing with the numbers {0, 1, 2, 3, ...}. Then, for our purposes, the "is equal to" relation is precisely given by the set {(0,0), (1,1), (2,2), (3,3), ...}.
When we say 5 = 5, that means that (5,5) is in the relation "is equal to". When we say , that means (5,6) is not in the relation "is equal to".
Another way to say this is that "5 is related to 5 by the 'is equal to' relation."
So for a pair of sets like A = {1, 2, 3, 4} and B = {a, b, c, d}, we can make up relations simply by listing ordered pairs (any pairs we choose), in which the first element is from the first set, and the second element is from the second set.
An example is C = {(1, b), (1, c), (2, a), (3,d), (4, d)}.
(This whole definition is a little watered down. Here the relation is defined as C, but the "real" definition of relation would be the ordered triple (A, B, C). I doubt your teacher would expect you to know this.)
An example of a function that you might have experience with is the square root function. What's special about functions is that given some input, there is a unique value associated with it by the function.
Suppose we're just working with the numbers {0, 1, 2, 3, ...}. Then the square root function will only be defined for the input values {0, 1, 4, 9, 16, ...}, which is the domain, and the function can be described by the set {(0,0), (1,1), (4,2), (9,3), (16,4), ...}.
Note that the set C above is not a function because the value 1 is associated with 2 values, b and c. For it to be a function, each value in A must be associated with exactly 1 value in B.
Slope is a whole other concept, and since this post is getting long, I'll just leave it at that for now. If you'd like me to explain slope, I can, but maybe you don't like or understand my explanation for relations and functions, so one thing at a time.