Understanding Functions: Puzzled by f(x) = sqrt(x)?

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In summary, a function is defined as each element in the domain being paired with just one element in the range. This definition may seem contradictory for functions like f(x) = sqrt(x), which has two roots (+ and -), but in set theory, f(x) = sqrt(x) is not considered a function due to having more than one value for x > 0. However, f(x) = x^2 does fit the definition of a function. It is important to specify the domain and superset of the range when determining if something is a function.
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EvLer
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I am going on my own through a chapter on functions and here is something that puzzled me in the definition:

Each element in the domain is paired with just one element in the range.

I guess my calculus knowledge interferes, but what about function like f(x) = sqrt(x). It has two roots: + and -. How does set theory account for that? Or is sqrt(x) not a function in set-theoretic terms :confused:
Although f(x) = x^2 fits the definition of the function.
Could someone please explain?

Thanks in advance.
 
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  • #2
[itex]f(x)=\pm\sqrt{x}[/itex] is not a function, since f(x) has more than one value for all x > 0. However, [itex]f(x)=\sqrt{x}[/itex] is a function; [itex]\sqrt{x}[/itex] always refers to the positive square root.
 
  • #3
Yes, just look at the definition of function. If something doesn't meet the requirements of the definition, it just simply isn't a function.
But how can you tell whether something is a function until you specify its domain and the superset of its range? [itex]\sqrt{x}[/itex] isn't a function from N to N.
 
  • #4
Thanks :smile:
 

Related to Understanding Functions: Puzzled by f(x) = sqrt(x)?

1. What is the meaning of f(x) in a function?

The notation f(x) represents a function, where x is the input or independent variable and f(x) is the output or dependent variable. It shows the relationship between the input and output values.

2. What does sqrt(x) mean in a function?

The notation sqrt(x) represents the square root function, where x is the input value and the output value is the square root of x. For example, sqrt(4) = 2 since 2 is the square root of 4.

3. How do I read and interpret a function with f(x) = sqrt(x)?

The function f(x) = sqrt(x) can be read as "f of x equals the square root of x." This means that for any input value x, the output value will be the square root of x. For example, f(9) = sqrt(9) = 3, so when x is 9, the function output is 3.

4. What does the graph of f(x) = sqrt(x) look like?

The graph of f(x) = sqrt(x) is a curve that starts at the origin and increases as x increases. It is a half-parabola shape that approaches but never touches the x-axis. The graph is in the first quadrant only, as the square root of a negative number is undefined.

5. How do I solve equations involving f(x) = sqrt(x)?

To solve equations involving f(x) = sqrt(x), you need to isolate the f(x) on one side and the other terms on the other side. Then, you can square both sides to eliminate the square root and solve for x. It is important to check for extraneous solutions, as squaring both sides can introduce additional solutions that may not be valid for the original equation.

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