Is the function y = x + 2 one-to-one?

  • MHB
  • Thread starter mathdad
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In summary, a function is one-to-one when every y-value in the range is generated by a single, unique input x-value in the domain. This means that for every x-value, there is a different, unique answer for y. This function, y = x + 2, is one-to-one and linear. However, it is important to also consider the graph of a function to confirm if it is one-to-one or not. And an expression is not a function when its graph does not pass the vertical line test.
  • #1
mathdad
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Why is y = x + 2 one-to-one?
 
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  • #3
No matter what x is the value of y will be a different answer.

y = x + 2

Let x = 0

y = 0 + 2

y = 2

(0, 2)

Let x = 3

y = 3 + 2

y = 5

(3, 5)

Yes, for every x value, y has a different, unique answer.

So, it is one-to-one. It is also linear because the biggest power is 1.

Correct?
 
  • #4
fyi, using a few example function calculations does not confirm that a function is 1-1.

e.g. if you had "tested" using the same x-values with the function $y=\sqrt{9-x^2}$, you would have drawn an incorrect conclusion.

Having an idea of what the function graph looks like is important in determining things like domain, range, and whether or not a function is 1-1.

the function $y=\sqrt{9-x^2}$ is a semicircle of radius 3 in quadrants I and II ... it does not pass the horizontal line test.
 
  • #5
It is important to have a picture in mind of the basic functions to help determine if it is one-to-one or not.

When is an expression not a function?
 
  • #6
RTCNTC said:
It is important to have a picture in mind of the basic functions to help determine if it is one-to-one or not.

When is an expression not a function?

come on, you know this ... when its graph doesn't pass the vertical line test
 
  • #7
Good information.
 

FAQ: Is the function y = x + 2 one-to-one?

Why is it important to determine if a function is one-to-one?

Determining if a function is one-to-one is important because it tells us whether there is a unique output for every input. This means that the function is injective, and each input has a corresponding output without any repetition. If a function is not one-to-one, it is not injective, and there can be multiple outputs for the same input, making it difficult to determine a clear relationship between the variables.

What does it mean for a function to be one-to-one?

A function is one-to-one when each input has a unique output. This means that no two different inputs can have the same output. Visually, a one-to-one function has a distinct curve that does not intersect with itself, indicating that each input has a unique output.

How can you determine if a function is one-to-one algebraically?

To determine if a function is one-to-one algebraically, we can use the horizontal line test. We draw a horizontal line across the function's graph, and if the line intersects the graph at more than one point, the function is not one-to-one. Additionally, we can also use the vertical line test; if a vertical line drawn on the graph intersects the function at more than one point, it is not one-to-one.

Can a function be one-to-one if it is not a straight line?

Yes, a function can be one-to-one even if it is not a straight line. For example, the function y = x² is one-to-one because each input has a unique output, and the graph does not intersect with itself. On the other hand, a straight line function like y = 2x is not one-to-one because there are multiple inputs that give the same output.

Why is y = x + 2 one-to-one?

Y = x + 2 is one-to-one because it passes the horizontal and vertical line tests. When we draw a horizontal line across the graph, it only intersects at one point, and when we draw a vertical line, it only intersects at one point as well. This means that each input has a unique output, satisfying the definition of a one-to-one function.

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