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Chapter 2: Problem 62

Determine whether each relation defines \(y\) as a function of \(x\). \(y=\sqrt{x-7}\)

Short Answer

Expert verified
Yes, \( y = \sqrt{x - 7} \) defines \( y \) as a function of \( x \).

Step by step solution

01

Understanding the Problem

Determine if the given equation \( y = \sqrt{x - 7} \) defines \( y \) as a function of \( x \). A relation defines \( y \) as a function of \( x \) if every \( x \)-value is associated with exactly one \( y \)-value.
02

Isolating Variables

Observe the equation \( y = \sqrt{x - 7} \). For each \( x \) value, calculate the corresponding \( y \) value. The expression inside the square root must be non-negative, so \( x - 7 \geq 0 \).
03

Finding the Domain

Solve \( x - 7 \geq 0 \) to find the domain of the function. Adding 7 to both sides, we get \( x \geq 7 \). This means for any \( x \) value that is 7 or greater, the expression \( \sqrt{x - 7} \) will be defined.
04

Analyzing the Square Root

Since the square root function always produces a single non-negative output for any non-negative input, for each \( x \geq 7 \), there is only one corresponding \( y \)-value given by \( y = \sqrt{x - 7} \).
05

Conclusion

Since every \( x \geq 7 \) corresponds to exactly one \( y \)-value, the relation \( y = \sqrt{x - 7} \) does define \( y \) as a function of \( x \).

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Key Concepts

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

Domain of a Function
The domain of a function refers to all possible input values (usually represented by \( x \)) that make the function work and yield real number outputs. To determine the domain, you need to identify values of \( x \) that do not lead to undefined or non-real results.
In the context of the given problem, we have the equation \( y = \sqrt{x - 7} \). The square root function requires its argument to be non-negative. Therefore, you set up the inequality:
\( x - 7 \geq 0 \). This simplifies to \( x \geq 7 \).
Thus, the domain of the function \( y = \sqrt{x - 7} \) is \( x \geq 7 \). Only values in this range will ensure that the function outputs real numbers.
Square Root Function
The square root function is written as \( \sqrt{x} \) and it returns the value that, when squared, equals \( x \). This function is defined only for non-negative inputs because there are no real numbers that squared result in a negative number. Hence, for a function involving a square root like \( y = \sqrt{x - 7} \), the expression inside the square root must be non-negative:
- Ensure \( x - 7 \geq 0 \)
- Simplify to \( x \geq 7 \)
The output \( y \) of a square root function is always non-negative as well, meaning \( y \geq 0 \). Understanding these properties helps us determine the nature and validity of a function involving square roots.
Isolating Variables
Isolating variables is a method to solve equations and determine specific values for variables. It involves manipulating the equation to get the variable of interest on one side of the equation.
In this problem, we started with \( y = \sqrt{x - 7} \) and needed to determine the possible values for \( x \) by setting the expression inside the square root to be non-negative:
- Start with \( x - 7 \geq 0 \)
- Add 7 to both sides to isolate \( x \)
- This gives \( x \geq 7 \)
This process helps in finding the domain, as it shows the range of values for \( x \) where the function will be defined.

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