How to we do the inverse of y=(x-1)^2 ?

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In summary, to find the inverse of y=(x-1)^2, we switch the positions of x and y, solve for y, and put it in terms of the given variable if possible. However, for x=2, which is not a function, there is no inverse. The rule must also be bijective, meaning it is both injective and surjective, which y = x^3-x does not meet.
  • #1
cmab
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How to we do the inverse of y=(x-1)^2 ?

Would it be x = sqrt(y) +1 ?
 
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  • #2
Let's see.

[tex]y = (x-1)^2[/tex]

Now switching x and y we get

[tex]x = (y-1)^2[/tex]
[tex]\sqrt{x} = y - 1[/tex]
[tex]y = \sqrt{x} + 1[/tex]

When you find inverses, you usually want to put the inverse in terms of the given variable if possible. Sometimes you'll see, it is quite impossible.

Example: Find the inverse of [tex]y = x^3-x[/tex]

Jameson
 
  • #3
but in my problem, it must have respect to y...

How about the reciprocal of x=2, would it be y=2 ? Just swapping the variable.
 
  • #4


The issue with finding the inverse of x=2 is that x=2 isn't a function. A function is an ordered tuple from one set to another. x=2 only refers to an one element of one set. Furthermore, if you want your function to have an inverse the rule has to have other requirements. It has to be bijective, which means it has to be both injective and surjective. Hence [tex]y = x^3-x[/tex]
has no inverse since solving for x gives multiple functions.
 
  • #5


Yes, that is correct. To find the inverse of a function, we can switch the x and y variables and solve for y. In this case, we would have x = (y-1)^2 and then take the square root of both sides to get y = sqrt(x) + 1. This is the inverse function of y=(x-1)^2.
 

1. How do we find the inverse of a quadratic function?

The inverse of a quadratic function can be found by switching the x and y variables and solving for y. This will result in a new function that represents the inverse of the original quadratic function.

2. What is the general form of an inverse quadratic function?

The general form of an inverse quadratic function is y = √(x + a) + b, where a and b are constants. This is also known as the square root function.

3. Can all quadratic functions have an inverse?

No, not all quadratic functions have an inverse. For a quadratic function to have an inverse, it must pass the horizontal line test, meaning that every horizontal line intersects the function at most once.

4. How do we graph an inverse quadratic function?

To graph an inverse quadratic function, we can first graph the original quadratic function and then reflect it over the line y = x. This will give us the graph of the inverse function.

5. What is the relationship between a quadratic function and its inverse?

The inverse of a quadratic function is the reflection of the original function over the line y = x. This means that the x and y values of the inverse function are switched, but the overall shape of the function remains the same.

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