Square root function: x intercept paradox?

In summary, the conversation discusses finding the x intercept of a function y= sqrt(x+4)+1 and explains how the first attempt at solving for x using y=0 does not yield a real solution due to the square root function not having negative values. The second attempt at finding the x intercept by plugging in -3 for x results in a point (-3,2) which is accepted by the graph. The expert further explains that the function has been shifted upward by 1 unit, causing the original x-axis intercept to no longer exist.
  • #1
Dapperdub
7
0
Here's my equation y= sqrt (x+4)+1

I want to find the x intercept. Ok so i replace y with 0 and solve for x
0= √(x+4)+1
-1=√(×+4)
(-1)2=×+4
1=×+4
-3=×

So it looks like the x intercept is (-3,0)

But then when i go back and plug in -3 in place of x...
y=√(-3+4)+1
y=1+1
y=2

I now have a point (-3,2)!

I checked the graph and it accepts the point (-3,2). So what's wrong with my first work putting in 0 for y and giving me (-3,0)? Please help me understand
 
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  • #2
Dapperdub said:
-1=√(×+4)
(-1)2=×+4
Here is the problem. The square root cannot be negative so the first line does not have solutions.
By squaring you create additional solutions to the equation which are not real.
 
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  • #3
mfb said:
Here is the problem. The square root cannot be negative so the first line does not have solutions.
By squaring you create additional solutions to the equation which are not real.

Thx for the response!

So then, is it classified as an imaginary root?
 
  • #4
If you look for x intercepts, you probably look for real values. There are imaginary values of x where the function (seen as complex function) is zero, yes.
 
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  • #5
The thing with square root functions is whenever you attempt to solve for x, you end up turning them into conical parabolas. This means you can get x-intercepts for the conical parabola which don't actually exist for the square root function.
 
  • #6
Dapperdub said:
Here's my equation y= sqrt (x+4)+1

I want to find the x intercept. Ok so i replace y with 0 and solve for x
0= √(x+4)+1
-1=√(×+4)
(-1)2=×+4
1=×+4
-3=×

So it looks like the x intercept is (-3,0)

But then when i go back and plug in -3 in place of x...
y=√(-3+4)+1
y=1+1
y=2

I now have a point (-3,2)!

I checked the graph and it accepts the point (-3,2). So what's wrong with my first work putting in 0 for y and giving me (-3,0)? Please help me understand
You have the upper branch of a square root function, and it has been shifted UPWARD by 1 unit, and therefore what once was an x-axis intercept has been stopped; as well as any intersection with the x-axis.

(I made a drawing and saved, using "whiteboard" but the file is missing and is therefore not available to upload.)
 
Last edited:

Related to Square root function: x intercept paradox?

1. What is the square root function?

The square root function is a mathematical function that takes a number as input and returns the number that, when multiplied by itself, gives the input number. For example, the square root of 9 is 3 because 3 multiplied by 3 is 9.

2. What is the x-intercept paradox in the square root function?

The x-intercept paradox in the square root function refers to the fact that the square root of a negative number is not a real number. This means that when graphing the square root function, there is no x-intercept, even though the function technically crosses the x-axis at x=0.

3. Why does the square root function have an x-intercept paradox?

The square root function has an x-intercept paradox because the square root of a negative number results in an imaginary number. Imaginary numbers cannot be plotted on a traditional x-y plane, so the square root function appears to have no x-intercept even though it technically crosses the x-axis at x=0.

4. How do we graph the square root function with the x-intercept paradox?

To graph the square root function with the x-intercept paradox, we can use a complex plane, which includes both real and imaginary numbers. This allows us to plot the imaginary solutions of the square root function and see that it does in fact cross the x-axis at x=0.

5. Can the x-intercept paradox be solved?

No, the x-intercept paradox cannot be solved because the square root of a negative number will always result in an imaginary number. This is a fundamental property of the square root function and cannot be changed. However, using a complex plane, we can better understand and graph the square root function with the x-intercept paradox.

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