Explaining the Inverse of f(x)=sq.rt.(x+1) and f(x)=sq.rt.(x+10)

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The discussion focuses on finding the inverses of the functions f(x)=sqrt(x+1) and f(x)=sqrt(x+10), resulting in y=x^2-1 and y=x^2-10, respectively. The necessity of stating "x is greater than or equal to 0" for the inverses is explained by the fact that square root functions only produce non-negative outputs, thus limiting their range. The range of the original functions is [0, infinity), which becomes the domain of their inverses. When converting a function to its inverse, the domain and range are swapped, highlighting the relationship between the two. Understanding this concept is crucial for grasping how functions and their inverses interact.
AznBoi
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Ok here are some examples that I am confused with:

f(x)=sq.rt.(x+1)
Inverse: y=x^2-1, x greater or equal to 0??

f(x)=sq.rt.(x+10)
Inverse: y=x^2-10, x greater or equal to 0??

How come you need the "x is greater or equal to 0" after each inverse?Can someone explain to me why this is? Is it because you can only inverse the positive side of the sq.rt.? How come you can't inverse the other side? I just don't get why you need to right that in the answer. Thanks a lot! :smile:
 
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In your two expressions for the f's, what are their maximal range?
 
The range of sq.rts. are infinity. or y is greater or equal to 0. Is that why?
 
AznBoi said:
The range of sq.rts. are infinity. or y is greater or equal to 0. Is that why?
"The range of sqrt. are infinity", whatever does that mean?
 
The range of square roots are (0, infinity) on the positive side. max range= infinity?
 
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If you think of a function as a machine, where you put in an x, it does something to it, and shoots out a y, then all the possible y values make up the range. So you are right, the range of both is y\geq0 What happens to the range and domain to a function when it is converted to it's inverse? Take a look at the graphs it should become apparent. If not, look at the table of values.
 
hint: If you were to switch all the x and y values of a function, what would happen? The points (0.1) (3,2) and (8,3) all lie on the graph of y=\sqrt{x+1}. The points (1,0) (2,3) (3,8) all lie on the graph of y=x^2-1 See a pattern? What would that do to the domain and range?

Sorry for the premature post, I tried switching to advanced mode and accidentally pressed submit message.
 
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AznBoi said:
The range of square roots are (0, infinity) on the positive side. max range= infinity?
That's much better than "are infinity"! Although I would say [0, infinity), specifically including 0 as a possible value. The point is that, if f is a function from A to B, then f-1 is a function from B to A: domain and range are swapped. If the range of f is [0, infinity), then the domain of f-1 is [0, infinity).
 
thanks a lot :-p
 

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