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Chapter 2: Problem 11

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x^{2}+y=16$$

Short Answer

Expert verified
Yes, the equation \(x^{2}+y=16\) defines \(y\) as a function of \(x\).

Step by step solution

01

Rearrange the equation to isolate y

The equation is given as \(x^{2}+y=16\). If we rearrange the equation to solve for \(y\), it becomes \(y=16 - x^{2}\).
02

Determine whether for every x, there is exactly one unique y

To see if for every \(x\) there is exactly one \(y\), we can consider any number in place of \(x\). For whatever number we choose, we get a unique value of \(y\). Thus, each value of \(x\) results in exactly one unique value of \(y\).

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Key Concepts

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

Solving Equations
Solving equations is a fundamental skill in algebra that involves finding the value of unknown variables that satisfy the equation. In our exercise, we began with the equation \(x^2 + y = 16\).
To solve for \(y\), we perform algebraic manipulation to isolate it on one side of the equation.
The rearrangement involves moving the \(x^2\) term to the other side, which changes the equation into \(y = 16 - x^2\).
This manipulation helps in expressing \(y\) as a function of \(x\), where \(y\) is dependent on the value of \(x\).
Remember:
  • Always aim to isolate the variable of interest by using inverse operations.
  • Keep both sides of an equation balanced by doing the same operation on each side.
The process of solving equations is crucial, especially when identifying relationships between variables in functions.
Unique Solutions
A critical aspect in functions is determining if every input (\(x\)) yields a unique output (\(y\)). This means for each value of \(x\), there must be exactly one corresponding \(y\).
In our example, after rearranging the given equation to \(y = 16 - x^2\), we can see that for any chosen value of \(x\), there鈥檚 only one way to compute \(y\), making it a unique solution.
This characteristic confirms that \(y\) is indeed a function of \(x\).
Why unique solutions matter:
  • They ensure consistency and predictability in mathematical modeling.
  • They signify a well-defined relationship between variables, crucial in mathematics and real-world applications.
Thus, determining uniqueness is a cornerstone in verifying if an equation expresses a function.
Algebraic Manipulation
Algebraic manipulation involves using various algebraic rules and techniques to simplify, rearrange, or solve equations. Here, we transformed \(x^2 + y = 16\) into \(y = 16 - x^2\).
The process depended on our understanding of how to handle equations:
  • Identify the terms that need to be moved to isolate the desired variable.
  • Use inverse operations like subtraction, addition, multiplication, or division.
  • Be mindful of maintaining equality by applying operations to both sides of the equation.
By practicing algebraic manipulation, we develop the skills to work through more complex equations, solve them, and express relationships in simpler forms. This is particularly useful in calculus and other higher-level mathematics topics.

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Most popular questions from this chapter

Group members should consult an almanac, newspaper, magazine, or the Internet to find data that lie approximately on or near a straight line. Working by hand or using a graphing utility, construct a scatter plot for the data. If working by hand, draw a line that approximately fits the data and then write its equation. If using a graphing utility, obtain the equation of the regression line. Then use the equation of the line to make a prediction about what might happen in the future. Are there circumstances that might affect the accuracy of this prediction? List some of these circumstances.

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)=\sqrt{-x+1} $$

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)=-|x+3| $$

Describe one advantage of using \(f(x)\) rather than \(y\) in a function's equation. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. On the other hand, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ h(x)=(x-1)^{2}+2 $$
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