# Find range of functions

## 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)

## 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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eumyang
Homework Helper

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

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

However, in the function
$$f(x) = \sqrt{x}$$
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.

symbolipoint
Homework Helper
Gold Member

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.

HallsofIvy
Homework Helper

## 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)

## 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, $\sqrt{4- x^2}$ is defined as the positive number such that its square is $4- x^2$. Similarly, $\sqrt{4- x}$ is defined as the positive number whose square is 4- x.