/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Determine whether or not the equ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 45

Determine whether or not the equation represents \(y\) as a function of \(x\). $$ 2 x+3 y=4 $$

Short Answer

Expert verified
Yes, the equation represents \(y\) as a function of \(x\).

Step by step solution

01

Recall the definition of a function

A relation is a function if for every input value there is exactly one output value. In other words, if an equation represents a function, each value of \(x\) should correspond to one and only one value of \(y\).
02

Solve for y

Begin by solving the given equation for \(y\) in terms of \(x\):1. Start with the equation \(2x + 3y = 4\).2. Subtract \(2x\) from both sides to isolate terms involving \(y\): \[ 3y = 4 - 2x \]3. Divide every term by 3 to isolate \(y\): \[ y = \frac{4 - 2x}{3} \] Now the equation is expressed as \(y\) in terms of \(x\).
03

Analyze the equation to determine if it is a function

In the equation \(y = \frac{4 - 2x}{3}\), \(y\) is expressed as a unique, single-valued expression in terms of \(x\). For any given value of \(x\), there is only one corresponding value of \(y\). This confirms that the equation represents \(y\) 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.

Function Definition
Understanding what a function is can be crucial when working with linear equations. A function is a special type of relationship between inputs and outputs. Here's what makes it unique:
  • For every input value (often represented by \(x\)), there must be one and only one corresponding output value (represented by \(y\)).
  • This means that in a function, each value of \(x\) must produce a single \(y\) value.
To determine whether an equation, like \(2x + 3y = 4\), represents a function of \(x\), we need to express \(y\) explicitly in terms of \(x\) and check if for each \(x\), the \(y\) is unique. If it is, then the equation describes a function of \(x\).
Solving Equations
To understand how \(y\) depends on \(x\), we need to solve the equation \(2x + 3y = 4\) by expressing \(y\) in terms of \(x\). This involves manipulating the equation so that \(y\) is isolated on one side.Let's go through the steps:
  • Start with the given equation: \(2x + 3y = 4\).
  • Subtract \(2x\) from both sides to maintain equality, leading to: \(3y = 4 - 2x\).
  • To solve for \(y\), divide every term on the right-hand side by 3: \(y = \frac{4 - 2x}{3}\).
By solving the equation, we now have \(y\) expressed in terms of \(x\), making it easier to analyze the relationship and function.
Isolating Variables
Isolating a variable is a fundamental skill in algebra that helps to simplify equations and solve them. In the context of functions, isolating \(y\) when determining if an equation represents a function is crucial.Here's how we isolated \(y\) in the equation \(2x + 3y = 4\):
  • First, identify the term involving \(y\) which is \(3y\) in this case.
  • Next, separate it from other terms by subtracting \(2x\) from both sides: resulting in \(3y = 4 - 2x\).
  • Finally, divide every term by the coefficient of \(y\) (which is 3 here), yielding \(y = \frac{4 - 2x}{3}\).
Isolating \(y\) allows us to clearly see how \(y\) is a function of \(x\) by presenting a direct expression that correlates these two variables. This is essential in not only understanding the problem but also in confirming the presence of a function.

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Most popular questions from this chapter

In Exercises \(21-41,\) determine analytically if the following functions are even, odd or neither. $$ f(x)=7 $$

In Exercises \(1-12\), sketch the graph of the given function. State the domain of the function, identify any intercepts and test for symmetry. $$ f(x)=x^{3} $$

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=f(x)+3 $$

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=f(x+3) $$

In Exercises \(74-77,\) use your graphing calculator to approximate the local and absolute extrema of the given function. Approximate the intervals on which the function is increasing and those on which it is decreasing. Round your answers to two decimal places. $$ f(x)=x^{2 / 3}(x-4) $$
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