Algebra I
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CA Algebra I: Number Properties and Absolute Value
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Simple Logical Arguments
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CA Algebra I: Graphing Inequalities
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CA Algebra I: Slope and Y-intercept
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CA Algebra I: Systems of Inequalities
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Factoring Quadratics
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CA Algebra I: Completing the Square
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CA Algebra I: Quadratic Equation
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CA Algebra I: Quadratic Roots
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CA Algebra I: Rational Expressions 1
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CA Algebra I: Rational Expressions 2
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CA Algebra I: Word Problems
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CA Algebra I: More Word Problems
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CA Algebra I: Functions
CA Algebra I: Functions 79-80, functions, domain and range
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- All right, problem 79.
- What is the domain of the function shown
- on the graph below?
- That sounds like fancy word.
- But a function is just a rule.
- It's just a rule that maps from one set
- of numbers to another.
- And the set of numbers that it maps from is the domain.
- And the set of numbers that it maps to is the range.
- So if this is the set of numbers of the domain, this is
- set of numbers of the range, the function tells us how do
- we get from this number, some place in domain, to that
- number, some place in the range.
- So in general, the domain is all the numbers for which this
- mapping, or this function, is defined.
- OK, so let's see, so the domain is where is this
- function defined?
- And in general, when you plot it on the xy-axis, the domain
- is all of the x values that you're defined over and the
- range is all the y values.
- So the function is a mapping from x to y.
- So let's see where it's defined when x is equal to 4.
- It tells us we're mapping from x is equal to 4 to y is equal
- to negative 1.
- So it's defined for x is equal to 4, for x is equal to 5, for
- x is equal to 2, and it's defined for x is equal to 1.
- So it's defined for all of those numbers.
- That's the domain.
- This function is not defined.
- We don't know what happens when x is equal to minus 2.
- It's not defined.
- It doesn't tell us what y equals here.
- So these are the only numbers for which
- this function is defined.
- It tells us that 4 maps to negative 1, that 5 maps to
- negative 2, that 2 maps to minus 5, and that 1
- maps to minus 4.
- That's all it tells us.
- And this is actually the range.
- And that's the domain.
- So let's see, 4, 5, 2, 1.
- This is the domain.
- That's the set of all numbers for which
- this function is defined.
- This minus 1, 2, 5, and 4.
- This is the range right here.
- Choice B is the range.
- All right.
- I think we're on the last problem.
- This is going to be a lot of cutting and pasting.
- I don't know if it's going to fit.
- I have to shrink it.
- OK, they say which of the following is not a function?
- That's fascinating.
- OK, I copied it.
- And now I have pasted it.
- And it did all barely fit on this page.
- So which is not a function?
- And remember, what I said in the beginning.
- I don't have space below here.
- Let's think about what a function is and then we could
- look at the graph.
- So a function is a mapping from one set to another.
- From set A to set B.
- Where A is the domain and B is the range.
- And this is important thing to realize about a function.
- So if this is set A, and if this is set B, that for any
- value in set A, it only maps to one value in set B.
- You cannot have this.
- You can't have one value in set A going to this value and
- going to another value.
- So that would not make it a function.
- If we say x is the domain, if this is the set of all values
- we can put into x, and this all the set of values that can
- go into y or that can be y, then one value of x can't
- produce two values of y.
- You can have it the other way around.
- You could have two different values of x producing the same
- value of y.
- And we'll think about what that is.
- That's completely valid.
- You could have two numbers.
- When x is equal to 2 and x is equal to 7, you can still have
- y equaling the same thing.
- Say this is x is equal to 5.
- X is equal to 5 could be y is equal to 7.
- Or x is equal to 5 could be y is equal to 8.
- We don't know that.
- So let's see.
- Let's think about how we can visually think about this.
- For every value of x, we only have one definition for y.
- And people often call that the vertical line test. At no
- point on this graph can I draw a vertical line and it
- intersects it twice.
- If a vertical intersects it twice, that means for a given
- value of x, this function defines two different y's.
- Not going to happen.
- Now this function does for any given y have
- two different x's.
- Like that point and that point.
- That's OK.
- That's completely fine.
- So for example, when x is equal to minus 1.5 and x is
- equal to 3, they point to the same number.
- That's completely OK.
- As long as one of these doesn't point to
- two separate numbers.
- So that magenta is what that first graph is doing.
- Same thing here, for any given y we can have two
- x's pointing to it.
- And they're just fine.
- That's completely fine for a function.
- Now, we talk about function inverses, well I don't want to
- get too complicated.
- That's completely fine.
- If we do the vertical line test, at no point can we draw
- vertical line and get, for any given x, two separate y's.
- Same thing for here.
- Actually here, we have a complete 1:1 mapping.
- For any given x, there's one y.
- I think here it should be clear, that if I were to draw
- vertical line, let's say right there, for x is equal to 1, it
- has two definitions.
- Even better, for x is equal to 0, y could be
- minus 2 or plus 2.
- So that's the case that we said.
- For x is equal to 0, it's pointing to plus 2 and it's
- pointing to minus 2.
- It's pointing to two different values which you cannot have
- in a function.
- For a function, for any given x, you can only
- define one y value.
- So this relation is not a function.
- D.
- Next question.
- I think this is the last. It's a page on its own.
- Oh, that was the last question.
- Did I skip a problem?
- See that was 80.
- That's it.
- We're all done.
- All right, see you soon.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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