Why Are Negative Values Excluded in the Range of These Square Root Functions?

In summary: So, the negative values are not in the range because they would violate the definition of the function.
  • #1
Michael_Light
113
0

Homework Statement



Find the range of each of the following functions. All the functions are defined for the largest possible domain of values of x.

a) f(x) = √(4-x^2) b) f(x) = √(4-x)


Homework Equations





The Attempt at a Solution



The answers given are a) 0 ≤ f(x) ≤ 2 b) f(x) ≥ 0 . But my answers are a) -2≤ f(x)≤2 b) All real numbers . Can anyone explain what i had done wrong? Why negative numbers are excluded?
 
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  • #2


Because you wouldn't have a function otherwise. Consider the basic square root function
[tex]f(x) = \sqrt{x}[/tex]

You're probably thinking that there are two square roots of a number,
[tex]\pm\sqrt{x}[/tex]

However, in the function
[tex]f(x) = \sqrt{x}[/tex]
if we allow both positive and negative values, you would end up with a single x-value paired with two function values (like (16, 4) and (16, -4)). That's not allowed in functions.

Your original problem works the same way. The negative values will not be in the range, because otherwise you wouldn't have functions anymore.
 
  • #3


Notice in exercise #b, if x>4, then the function has no Real value. Also, the square root will not be less than 0, meaning the function will be in range of greater or equal to 0than 0 but not less than 0.
 
  • #4


Michael_Light said:

Homework Statement



Find the range of each of the following functions. All the functions are defined for the largest possible domain of values of x.

a) f(x) = √(4-x^2) b) f(x) = √(4-x)


Homework Equations





The Attempt at a Solution



The answers given are a) 0 ≤ f(x) ≤ 2 b) f(x) ≥ 0 . But my answers are a) -2≤ f(x)≤2 b) All real numbers . Can anyone explain what i had done wrong? Why negative numbers are excluded?
Because, as eumyang said, [itex]\sqrt{4- x^2}[/itex] is defined as the positive number such that its square is [itex]4- x^2[/itex]. Similarly, [itex]\sqrt{4- x}[/itex] is defined as the positive number whose square is 4- x.
 

1. What does "find range of functions" mean?

"Find range of functions" refers to the process of determining all possible output values of a mathematical function. This is typically done by substituting different input values into the function and observing the corresponding output values. The range of functions is often represented as a set of numbers or a graph.

2. Why is it important to find the range of functions?

Finding the range of functions is important because it helps us understand the behavior and limitations of a mathematical function. It also allows us to determine if a function is one-to-one or onto, which has implications for its inverse function and applications in real-world problem solving.

3. What is the difference between domain and range of functions?

The domain of a function refers to the set of all possible input values, while the range refers to the set of all possible output values. In other words, the domain is the set of x-values and the range is the set of y-values. The domain and range are both important in understanding the behavior of a function.

4. How do you find the range of a linear function?

The range of a linear function can be found by examining the slope and y-intercept of the function. If the slope is positive, the range is all real numbers greater than or equal to the y-intercept. If the slope is negative, the range is all real numbers less than or equal to the y-intercept. If the function is horizontal, the range is just the y-intercept.

5. Can the range of a function ever be empty?

Yes, the range of a function can be empty if there are no output values for any input values. This can occur if the function is undefined for certain input values, or if the function is bounded and there are no input values within the bounds. However, it is more common for the range to contain at least one value, as most functions have a continuous set of output values.

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