Problem 14
Question
Determine whether each relation describes \(y\) as a function of \(x\) $$y=\sqrt{x+3}$$
Step-by-Step Solution
Verified Answer
The relation \(y = \sqrt{x + 3}\) describes y as a function of x, as it passes the vertical line test with each x-value in the domain \(x \geq -3\) having a unique non-negative y-value.
1Step 1: Identify the domain of the relation
The given relation is \(y = \sqrt{x + 3}\). Since we cannot take the square root of a negative number, the domain of this relation is all real numbers \(x\) such that \(x+3 \geq 0\). Solving for x, we find that the domain of this relation is \(x \geq -3\).
2Step 2: Analyze the relation for unique y-values
Now that we know the domain of the relation, let's analyze the equation to see if it passes the vertical line test. For each value of x in the domain, we must ensure that there is only one corresponding y-value.
Since the square root function only returns non-negative values, the range of the relation will always be non-negative as well. Therefore, for each x-value in the domain, there will be exactly one non-negative value of y. Hence, the relation passes the vertical line test.
3Step 3: Conclusion
Since the relation passes the vertical line test, we can conclude that the given relation, \(y = \sqrt{x + 3}\), describes y as a function of x.
Key Concepts
Domain and RangeVertical Line TestSquare Root Function
Domain and Range
When we talk about the domain of a function, we are referring to all the possible inputs or x-values that the function can accept without resulting in undefined expressions. In the relation \( y = \sqrt{x + 3} \), we need to determine the values of \( x \) for which the square root expression is defined. Since the square root function is undefined for negative numbers, the expression \( x + 3 \) must be zero or positive.
Solving the inequality \( x + 3 \geq 0 \), we find that \( x \geq -3 \). Thus, the domain is all real numbers greater than or equal to -3.
The range of a function is concerned with all possible outputs or y-values the function can produce. Due to the characteristics of a square root, which only outputs non-negative values, the range for this function will be \( y \geq 0 \). Hence, for every x-value in the domain, the function returns a non-negative y-value, complying with the square root's definition.
Solving the inequality \( x + 3 \geq 0 \), we find that \( x \geq -3 \). Thus, the domain is all real numbers greater than or equal to -3.
- The domain is \( x \geq -3 \).
- It ensures the square root remains real-valued.
The range of a function is concerned with all possible outputs or y-values the function can produce. Due to the characteristics of a square root, which only outputs non-negative values, the range for this function will be \( y \geq 0 \). Hence, for every x-value in the domain, the function returns a non-negative y-value, complying with the square root's definition.
Vertical Line Test
The vertical line test is a simple visual method used to determine whether a relation is a function. It involves drawing vertical lines through the graph of the relation. If any vertical line crosses the graph at more than one point, the relation is not a function, because a single x-value would correspond to multiple y-values.
In the case of \( y = \sqrt{x + 3} \), when plotted, the graph represents only one y-value for every x-value in the domain. This is indicative of a function. A vertical line can cross the graph at most once for each x in the domain:
This means that the relation satisfies the vertical line test criteria, confirming that \( y = \sqrt{x + 3} \) is indeed a function.
In the case of \( y = \sqrt{x + 3} \), when plotted, the graph represents only one y-value for every x-value in the domain. This is indicative of a function. A vertical line can cross the graph at most once for each x in the domain:
- Every x-value matches precisely one y-value.
- Square root functions only output one value for any given input.
This means that the relation satisfies the vertical line test criteria, confirming that \( y = \sqrt{x + 3} \) is indeed a function.
Square Root Function
The square root function is an important type of function that often arises in mathematical problems. The basic square root function is \( y = \sqrt{x} \) but it can be altered with constant shifts or stretches as seen in \( y = \sqrt{x + 3} \). Understanding how modifications alter the graph's position is crucial.
Square root functions always return non-negative results as they represent a principal square root. This characteristic affects the function's domain and range. In the given problem, understanding these shifts and domains is key to interpreting how functions behave graphically and numerically.
- Adding a number inside the root like \( x + c \) shifts the graph horizontally.
- In \( y = \sqrt{x + 3} \), the graph shifts 3 units to the left, starting at \( x = -3 \).
Square root functions always return non-negative results as they represent a principal square root. This characteristic affects the function's domain and range. In the given problem, understanding these shifts and domains is key to interpreting how functions behave graphically and numerically.
Other exercises in this chapter
Problem 14
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