Problem 85 - College Algebra with Modeling and Visualization

Question

Determine if \(y\) is a function of \(x\). $$ \sqrt{x+1}=y $$

Step-by-Step Solution

Answer

Yes, \(y\) is a function of \(x\).

Step 1: Understand the Definition of a Function

A relation is a function if every input has exactly one output. Here, we need to determine if, for every value of \(x\), there is only one corresponding \(y\) value.

Step 2: Analyze the Given Equation

The equation given is \( \sqrt{x+1} = y \). To see if \( y \) is a function of \( x \), consider what happens for different \( x \) values.

Step 3: Consider the Domain of the Function

The expression \( \sqrt{x+1} \) is defined for \( x + 1 \geq 0 \) or \( x \geq -1 \). This means we only consider \( x \) values that are greater than or equal to -1.

Step 4: Check for Uniqueness of \( y \)

Since \( \sqrt{x+1} \) always gives a non-negative result (because the square root of a non-negative number is non-negative), for each \( x \), there is exactly one \( y \). This ensures the uniqueness of \( y \) for every \( x \) in the domain. For example, if \( x = 3 \), then \( \sqrt{3+1} = \sqrt{4} = 2 \).

Step 5: Conclude Whether y is a Function of x

Since for each \( x \) in the domain \( x \geq -1 \), there is exactly one \( y \), the relation described by the equation \( \sqrt{x+1} = y \) is indeed a function.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we mean the set of all possible input values that the function can accept. For the expression \( \sqrt{x+1} \), we have to ensure that the value inside the square root is not negative. This is because the square root of a negative number is not defined in the set of real numbers.

Let's look at what this means for our equation. The condition to keep the inside of the square root non-negative is:

• \( x + 1 \geq 0 \)
• This can be rearranged to \( x \geq -1 \)

This tells us that the domain of the function \( \sqrt{x+1} = y \) is all values of \( x \) that are greater than or equal to -1. So, anytime you're asked about the domain of such a function, think about where the inside of the root is non-negative!

Uniqueness of Outputs

In a function, each input must correspond to exactly one output. This is what makes a relation a true function. To determine uniqueness for \( y = \sqrt{x+1} \), we analyze whether each \( x \) will give us only one \( y \) value.

For our equation:

• Whenever \( x \) is substituted into the equation, \( x + 1 \) is always a non-negative number in the domain \( x \geq -1 \).
• The result \( \sqrt{x+1} \) is always non-negative and gives only one \( y \) for each \( x \).
• This means that for every value of \( x \) in the domain, \( y \) is unique.

For instance, if \( x = 3 \), then \( y = \sqrt{3+1} = \sqrt{4} = 2 \). Thus, there is no ambiguity or multiple outputs for a single input \( x \), confirming the uniqueness of the output.

Square Root Function

The square root function is a type of function that involves finding a number which, when multiplied by itself, gives the original number under the root. Its general form is \( \sqrt{x} \).

Square root functions have these key characteristics:

• The output is the principal square root, meaning it considers only the non-negative result.
• For \( \sqrt{x+1} \), it translates the basic square root function by 1 unit to the left (since the inside is \( x+1 \)).
• These functions are naturally restricted to non-negative outputs to maintain real number values.

Understanding square root functions can be tricky, but always remember that they only consider the principal (positive) root, which simplifies checking for uniqueness of outputs. So, whenever encountering a square root function, consider these properties to analyze functions more effectively.