Please! I'm taking an online Algebra II course, and it's frustrating. Can you include examples too? Please(: I'm behind as it is.
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I keep this function madness straight by thinking of mail delivery.
A house can get multiple letters, a house can get no letters. These mappings are ok in a function.
If a letter has to be torn apart and the pieces delivered to multiple houses, that's not a function.
In the pictures, there is a list of stuff on the left called the domain. Those are the letters.
The list on the right is the range. Those are the houses.
Make sure there are no mappings assigning a letter to multiple houses and you'll be ok.
When you're given a list of coordinates and asked if they make a function, the first coordinates make the domain (letters), the second make the range (houses). Set up a chart and check if any letters have to be torn apart and delivered to multiple houses.
(3, 7), (4, 7) can be points on the graph of a function: 3 and 4 are delivered to house 7. That's ok.
(3, 2), (3, 7) cannot be points on the graph of a function: The 3 is addressed to both houses 2 and 7 and cannot be delivered intact.
For one-to-one functions, you want each house getting only one letter.
A house can get multiple letters, a house can get no letters. These mappings are ok in a function.
If a letter has to be torn apart and the pieces delivered to multiple houses, that's not a function.
In the pictures, there is a list of stuff on the left called the domain. Those are the letters.
The list on the right is the range. Those are the houses.
Make sure there are no mappings assigning a letter to multiple houses and you'll be ok.
When you're given a list of coordinates and asked if they make a function, the first coordinates make the domain (letters), the second make the range (houses). Set up a chart and check if any letters have to be torn apart and delivered to multiple houses.
(3, 7), (4, 7) can be points on the graph of a function: 3 and 4 are delivered to house 7. That's ok.
(3, 2), (3, 7) cannot be points on the graph of a function: The 3 is addressed to both houses 2 and 7 and cannot be delivered intact.
For one-to-one functions, you want each house getting only one letter.
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A function is a relation (a set of ordered pairs) for which there is only one value in the range for each value in the domain. Said another way, a function has one y for each x. If you look at a graph, a function will pass the vertical line test.
A one-to-one function is a function that has only one value in the domain for each value in the range. Said another way, a one-to-one function is a function that has one x for each y. If you look at a graph, a one-to-one function will pass the vertical line test AND the horizontal line test.
For example, look at the graph of the parabola y=x². If you draw a vertical line through it at say, x=4, there will only be one y-value for the x-value where you've drawn the line. However, if you draw a horizontal line through it at, say, y=4, you will see that there are two x-values for that y-value. These are at the points (-2, 4) and (2, 4). While this is a function, it is not one-to-one.
On the other hand, look at the line y=3x+2. A vertical line will only go through one point no matter which x-value you choose. In addition, a horizontal line will only go through one point on the graph no matter which y-value you choose. You can therefore see that this is a one-to-one function.
One reason one-to-one functions are special is that their inverses are also functions. Looking at the two examples above, switch the x and y then solve for y (which is how to find an inverse).
In the first example, you get y=±√x as the inverse. If you were to plug in 9 for x you would get two answers (+3 and -3). This inverse is not a function.
In the second example you get y=(x-2)/3 as the inverse. This is also a line, which is a function.
A one-to-one function is a function that has only one value in the domain for each value in the range. Said another way, a one-to-one function is a function that has one x for each y. If you look at a graph, a one-to-one function will pass the vertical line test AND the horizontal line test.
For example, look at the graph of the parabola y=x². If you draw a vertical line through it at say, x=4, there will only be one y-value for the x-value where you've drawn the line. However, if you draw a horizontal line through it at, say, y=4, you will see that there are two x-values for that y-value. These are at the points (-2, 4) and (2, 4). While this is a function, it is not one-to-one.
On the other hand, look at the line y=3x+2. A vertical line will only go through one point no matter which x-value you choose. In addition, a horizontal line will only go through one point on the graph no matter which y-value you choose. You can therefore see that this is a one-to-one function.
One reason one-to-one functions are special is that their inverses are also functions. Looking at the two examples above, switch the x and y then solve for y (which is how to find an inverse).
In the first example, you get y=±√x as the inverse. If you were to plug in 9 for x you would get two answers (+3 and -3). This inverse is not a function.
In the second example you get y=(x-2)/3 as the inverse. This is also a line, which is a function.