How is any equation of the form y= -x + b its own inverse?
write it as
x + y = b
now recall how you take the inverse.
Isn't your first step simply to switch around the x and y variables?
So what you you get ?
The same equation, just simply switched:
if you have
x+y= b, the inverse is y+x=b, which is the same thing and therefore its own inverse?
Thank you!
To understand how an equation of the form y = -x + b is its own inverse, let's first define what it means for a function or equation to be its own inverse.
An inverse function undoes the actions of the original function, so when you apply an inverse to a function, you should get back the original input. In the case of an equation, if you substitute the output value (y) as the input and get back the original input (x), then the equation is its own inverse.
Now let's see how to verify if the equation y = -x + b is its own inverse:
1. Start with the given equation: y = -x + b.
2. Replace y (the output value) with x (the original input) and solve for x.
- We substitute y = x into the equation: x = -x + b.
- Add x to both sides of the equation: x + x = b.
- Simplify: 2x = b.
- Divide both sides of the equation by 2: x = b/2.
3. We obtained x = b/2, which suggests that the input value x is equal to half of the constant term b.
4. Now, substitute this value (x = b/2) back into the original equation y = -x + b, and check if we get the original output value y.
- Substituting x = b/2 into the equation gives: y = -(b/2) + b.
- Simplifying: y = (b - 2b) / 2 = (-b) / 2 = -b / 2.
5. Comparing the expression -b/2 with the original output value, y, we can see that they are equal. Thus, the equation y = -x + b is indeed its own inverse.
To summarize, by substituting y with x in the equation y = -x + b and solving for x, we obtained the original input value x. Substituting this value back into the equation confirmed that it reproduces the original output value y. Therefore, the equation y = -x + b is its own inverse.