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Chapter 2: Problem 20

Determine whether each equation defines y as a function of x. $$ y=-\sqrt{x+4} $$

Short Answer

Expert verified
Yes, the given equation \(y=-\sqrt{x+4}\) defines y as a function of x for \(x \geq -4\).

Step by step solution

01

Examine the equation

The given equation is \(y = -\sqrt{x+4}\). It involves a square root function, which are always positive for real numbers. In our equation, there's a minus (-) sign before the square root, hence, the resulting function will calculate the negative square root of \(x + 4\).
02

Determine the domain of x

A square root function is only valid for values of the input greater than or equal to zero. This means that for \(x+4\), we can say \(x+4 \geq 0\). Solving this gives \(x \geq -4\). This is our domain, the set of all possible values of x.
03

Check if each value of x gives a unique y

For each value of x in the domain (i.e., each x 鈮� -4), the equation provides a unique y value because the square root function is a well-defined function. By inserting different values of x from the domain into the equation, it can be seen each x gives a unique y, either positive or negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of a function
When discussing functions, the domain refers to all possible input values (usually "x" values) that the function can accept. In the given example of the equation \(y = -\sqrt{x+4}\), determining the domain starts by considering any restrictions imposed by the function. For square roots, we can only take the square root of non-negative numbers, meaning the expression inside the square root (in this case, \(x+4\)) must be greater than or equal to zero.

So, for \(x+4 \geq 0\), solving gives \(x \geq -4\). This means the function's domain is all real numbers greater than or equal to -4.
  • A square root function restricts the domain to values where the expression inside is non-negative.
  • The domain is the set of allowable inputs that keep the function "valid."
  • For \(y = -\sqrt{x+4}\), the input must be \(x \geq -4\).
Square root function
The square root function is a common type of function that involves taking the square root of some expression. When analyzing the equation \(y = -\sqrt{x+4}\), you've got a square root function, where the expression inside the square root dictates specific behavior for the function. The appearance of a square root usually ensures that the output is always non-negative because square roots of non-negative numbers are non-negative themselves.

However, our equation has a negative sign in front of the square root, flipping all positive values to negative ones. This is crucial: it means instead of typically positive output, we have negative output.
  • Square roots simplify to only non-negative numbers when without other modifications.
  • The negative sign usually changes positive square root results into their negative counterparts.
  • This flips the common "positive-only" structure of square roots to a negative-only one.
Unique y-values
The idea of unique y-values means each x-value within the domain results in exactly one corresponding y-value. For a function, each input should map to a single output, reinforcing the definition of a function. In the context of \(y = -\sqrt{x+4}\), each x-value greater than or equal to -4 passes through the transformation 鈥� insertion through the formula 鈥� and results in only one y-value.

This establishes it as a well-defined function. No matter which x you choose from the domain, it can't map to more than one y due to the nature of square root calculations.
  • Functions map each input to a unique output.
  • Unique y-values ensure we don't have multiple outputs for a single input.
  • The well-behaved nature of square roots 鈥� ensuring a consistent result 鈥� maintains y-values' uniqueness.

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Most popular questions from this chapter

Consider the function defined by $$ \\{(-2,4),(-1,1),(1,1),(2,4)\\} $$ Reverse the components of each ordered pair and write the resulting relation. Is this relation a function?

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