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Chapter 1: Problem 13

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 isolating y

We start by rearranging the given equation, with the end goal of expressing \(y\) in terms of \(x\). To do this, the process is to subtract \(x^2\) from both sides of the equation, resulting in \(y=16-x^2\)
02

Analyze the result

Observing the equation \(y=16-x^2\), we can see that for any given \(x\) value, there will be exactly one associated \(y\) value. This output results from subtracting the square of \(x\) from 16, meaning there is only one possible \(y\) value.
03

Conclusion

Given our analysis in Step 2, we can therefore conclude that the equation \(y=16-x^2\) does indeed define \(y\) as a function of \(x\), as there is one unique \(y\) value for every \(x\) value.

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Key Concepts

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

Defining Functions
In precalculus, a function is a special type of relationship between sets where each input, often denoted as x, is related to exactly one output, usually denoted as y. This is summarized by the concept of function notation, such as f(x), which represents the output in terms of the input x.

Function definition is integral when working with equations. It involves finding an explicit expression for y in terms of x. To define a function, we typically rearrange an equation to solve for y. In the given exercise, by manipulating the equation to isolate y, we obtain an expression that represents y as a function of x, which in this case, is a quadratic expression.
Solving Equations
Solving equations is a cornerstone of algebra and precalculus. It involves finding the value(s) of the variable(s) that make an equation true. When equations involve functions, we aim to express one variable in terms of another to understand their relationship fully. For instance, when looking at an equation like \(x^2 + y = 16\), solving for y in terms of x helps us to visualize how changes in the value of x affect y. This is a stepping stone for further analysis in topics such as function analysis and graphing.
Function Analysis
Function analysis delves deep into understanding the nature of functions. It looks at properties such as domain and range, intercepts, intervals of increase and decrease, maxima and minima, and even broader behavior, like end-behavior and symmetry.

Analysing the function from our exercise, \(y = 16 - x^2\), involves looking at how the function behaves for different values of x. For every value of x, there is a unique corresponding y value, identifying the function's domain, which is all real numbers, and its range, which is y 鈮� 16. Such an analysis gives more context to the numbers and unveils the function's behavior.
Quadratic Functions
Quadratic functions are second-degree polynomials with the general form \(y = ax^2 + bx + c\), where a, b, and c are constants, and a is not zero. The graph of a quadratic function is a parabola which can open upward or downward depending on the sign of a.

Our exercise presented us with a specific type of quadratic function: \(y = 16 - x^2\). This function鈥檚 graph is a downward-opening parabola with a vertex at the maximum point (0,16). Quadratic functions are notable for their predictable symmetry and are often used in physics, engineering, and economics to model a variety of phenomena.

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Most popular questions from this chapter

In Exercises \(67-70,\) graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned} (x-3)^{2}+(y+1)^{2} &=9 \\ y &=x-1 \end{aligned}$$

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+3 x-2 y-1=0$$

What must be done to a function's equation so that its graph is shifted vertically upward?

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}-2 x+y^{2}-15=0$$

Describe a procedure for finding \((f \circ g)(x) .\) What is the name of this function?
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