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Chapter 1: Problem 11

Determine whether each equation defines y as a function of \(x .\) $$x+y=16$$

Short Answer

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

Step by step solution

01

Understand what the problem is asking

We need to determine if y can be expressed as a function of x in the equation \(x + y = 16\). A function means that for each input (in this case each value of x), there's only one corresponding output (value of y).
02

Isolate y in terms of x

Rearrange the equation to express y in terms of x. To do this, subtract x from both sides of the equation to isolate y on one side. Doing so gives us the equation \(y = 16 - x\).
03

Evaluate whether y is a function of x

Looking at the equation \(y = 16 - x\), we can see that for any given x, there is exactly one corresponding value for y. Thus, y is a function of x in this equation.

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Key Concepts

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

Understanding Function Notation
Function notation is a way to name functions in mathematics, making them clear and easy to use. It tells us that one variable (say, \(y\)) is changing in relation to another variable (like \(x\)). In other words, it emphasizes the role of variables in equations. For example, when you see \(f(x)\), you read it as "\(f\) of \(x\)." This means that \(f\) is the name of the function, and \(x\) is the input variable.

Using function notation is useful because:
  • It shows the relationship between variables clearly.
  • It helps in identifying which variable is dependent (usually \(y\)) and which is independent (usually \(x\)).
  • It allows us to easily substitute values into the function.
Converting equations like \(x + y = 16\) into function notation (\(y = 16 - x\)) helps us see that \(y\) is indeed a function of \(x\). There鈥檚 only one outcome for \(y\) for each \(x\).
Fundamentals of Linear Equations
Linear equations are equations that create straight lines when graphed. These equations usually take the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope \(m\) represents the rate of change, and the y-intercept \(b\) is the point where the line crosses the y-axis.

Key characteristics of linear equations:
  • They have a constant rate of change (the slope).
  • The graph of a linear equation is a straight line.
  • They can be written in different forms, such as slope-intercept form, standard form, or point-slope form.
In the equation \(x + y = 16\), if we rewrite it as \(y = 16 - x\), we can see that it's a linear equation. The slope here is \(-1\), meaning the line decreases by 1 unit of \(y\) for every 1 unit increase in \(x\).
Solving Equations for a Variable
Solving equations often involves isolating a variable to find its value given certain conditions. This process involves performing operations such as addition, subtraction, multiplication, or division on both sides of an equation to maintain equality. The goal is to have the variable by itself on one side of the equation, which allows us to express it as a function of another variable.

Here鈥檚 how you might solve an equation:
  • Identify the variable to solve for, such as \(y\) in \(x + y = 16\).
  • Perform operations to isolate the variable on one side. For example, subtract \(x\) from both sides to get \(y = 16 - x\).
  • Ensure that each operation is performed on both sides to keep the equation balanced.
By isolating \(y\), we can clearly see its relationship with \(x\), confirming that \(y = 16 - x\) gives precisely one value of \(y\) for each \(x\), showing \(y\) is a function of \(x\).

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Most popular questions from this chapter

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by [-5,5,1] viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$g(x)=\left|4-x^{2}\right|$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned}(x-3)^{2}+(y+1)^{2} &=9 \\\y &=x-1\end{aligned}$$

$$\text { Solve for } y: \quad x=y^{2}-1, y \geq 0$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The function \(f(x)=5\) is one-to-one.
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