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Chapter 2: Problem 64

Determine whether each relation defines \(y\) as a function of \(x\). \(y=\sqrt{2 x+9}\)

Short Answer

Expert verified
Yes, the relation defines y as a function of x.

Step by step solution

01

Understand the Definition of a Function

First, recall that a relation defines a function if each input value (typically represented by the variable x) is associated with exactly one output value (typically represented by the variable y).
02

Analyze the Given Relation

Examine the relation given: \[ y = \sqrt{2x+9} \] This is an equation involving the variables x and y.
03

Check for Each Input (x) Mapping to One Output (y)

Analyze if for every x there is only one y value. Because the square root function \( \sqrt{2x+9} \) yields only the positive root (as per the principal square root definition), each x-value results in a unique non-negative y-value.
04

Conclude if it is a Function

Since for every x-value in the domain of the function (such that the expression inside the square root is non-negative, i.e., \(2x + 9 \geq 0 \)), we indeed get a unique y-value. Hence, the given relation defines 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.

Definition of a Function
In algebra, a function is a relation where every input value (x) maps to exactly one output value (y). Imagine you're assigning each person in a classroom a unique ID number. If two people share the same ID number, it causes confusion. Similarly, in a function, each x-value must pair with one and only one y-value. This ensures clarity and consistency in mathematical relations. Therefore, for an equation to define a function, each input must have a single, unique output.
To determine if an equation represents a function, check if any x-value could produce more than one y-value. If that happens, it's not a function.
Principal Square Root
The principal square root is key to understanding certain functions. When you see a square root symbol, it generally refers to the principal square root, which is the non-negative root of a number. For example, the principal square root of 9 is 3, not -3, even though both squared give 9.
Consider the equation \( y = \sqrt{2x + 9} \). Here, the square root symbol automatically refers to the principal square root. This means \( y \) will always be non-negative for any valid \( x \). The principal square root helps ensure that for each x-value, there's only one possible y-value that is non-negative. This uniqueness is crucial in defining a function.
Unique Output Values
A defining characteristic of a function is that every input has a unique output. Let's look closely at the equation \ y = \sqrt{2x + 9} \. Because we are using the principal square root, each x-value gives only one non-negative y-value.
For example, if \(x = 0 \, \ \ y = \sqrt{2 \cdot 0 + 9} = \sqrt{9} = 3 \). There's no other y-value that can correspond to \(x = 0 \).
  • This consistency means that \ y \ is unique for every \ x \, which satisfies the criterion for being a function.
  • This uniqueness makes verifying functions easier, especially with equations like \( y = \sqrt{2x + 9} \).
Ensuring each input maps to a single output is why this particular equation defines y as a function of x.

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Most popular questions from this chapter

Express each relation using a different form. (For example, if the given form is a set of ordered pairs, use a graph.) There is more than one correct way to do this. $$ \begin{array}{c|c} x & y \\ \hline-1 & -3 \\ \hline 0 & -1 \\ \hline 1 & 1 \\ \hline 3 & 3 \end{array} $$

Which of the following defines \(y\) as a linear function of \(x ?\) A. \(y=\frac{1}{4} x-\frac{5}{4}\) B. \(y=\frac{1}{x}\) C. \(y=x^{2}\) D. \(y=\sqrt{x}\)

A taxicab driver charges \(\$ 2.50\) per mile. (a) Fill in the table with the correct response for the price \(f(x)\) the driver charges for a trip of \(x\) miles. (b) The linear function that gives a rule for the amount charged is \(f(x)=\) (c) Graph this function for the domain \\{0,1,2,3\\} using the set of axes at the right. $$ \begin{array}{c|c} x & f(x) \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline \end{array} $$

Write an equation of the line passing through the given point and satisfying the given condition. Give the equation (a) in slope-intercept form and (b) in standard form. See Example 6. $$ (7,2) ; \text { parallel to } 3 x-y=8 $$

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Find the slope of segment \(A C\).
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