/*! 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 35 Determine whether \(y\) is a fun... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 35

Determine whether \(y\) is a function of \(x\). $$ x^{2}+y^{2}=4 $$

Short Answer

Expert verified
No, \(y\) is not a function of \(x\) in the given equation.

Step by step solution

01

Write the given equation

The given equation is: \(x^{2}+y^{2}=4\)
02

Rearrange the equation to solve for \(y\)

Next, rearrange the equation to solve it for \(y\). Subtract \(x^{2}\) from both sides of the equation. The rearranged equation is: \(y^{2}=4-x^{2}\). Thereafter, take square root on both sides, its gives two solutions for \(y\), \(y=\sqrt{4-x^{2}}\) and \(y=-\sqrt{4-x^{2}}\)
03

Determine if \(y\) is a function of \(x\)

For a relationship to be a function, every x value must have exactly one corresponding y value. Here for every x value, we have two y values, one from \(y=\sqrt{4-x^{2}}\) and one from \(y=-\sqrt{4-x^{2}}\). Hence, \(y\) is not 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.

Vertical Line Test
To determine if a graph represents a function, we use a simple method known as the Vertical Line Test. This test helps us check whether for each value of the independent variable, there is a unique value for the dependent variable.

If a vertical line can intersect a graph at more than one point at any given time, then the graph does not represent a function. This is because a single x value should not be associated with multiple y values in a function. The Vertical Line Test quickly assesses whether the graph of a relation is a function by looking for any place where a vertical line crosses the graph more than once.

In the context of the given equation, which is a circle, drawing vertical lines through the graph will intersect it at two points. Thus, it fails the Vertical Line Test, confirming that the graph does not represent a function.
Implicit Equation
An implicit equation is an equation where the dependent and independent variables are intermixed together in a way that does not allow one to express directly in terms of the other. Such equations differ from explicit equations, where you can clearly solve for one variable.In the provided exercise, the equation is given as \(x^2 + y^2 = 4\), which is an example of an implicit equation. This equation involves both \(x\) and \(y\) in a combined manner. It is not immediately possible to express \(y\) as a function of \(x\) without rearranging.

Implicit equations are quite common in representing complex shapes like circles or curves, where it might be difficult to isolate a specific dependent variable. This characterization requires the use of algebraic manipulation or other methods to analyze how the variables are related.
Circle Equation
The equation \(x^2 + y^2 = 4\) is a classic example of a circle equation in a standard form. A circle centered at the origin with a radius can typically be represented as \(x^2 + y^2 = r^2\), where \(r\) is the radius.

This specific equation represents a circle with a center at the point \((0,0)\) and a radius of 2, since \(4 = 2^2\).

Understanding circle equations is essential in analyzing geometric shapes in algebra. Circles as geometric objects always satisfy this form of equation, showcasing symmetry about the origin. Additionally, circles do not have a one-to-one correspondence between \(x\) and \(y\), which often means they don't fulfill the functional definition, redirecting us back to concepts like the Vertical Line Test to determine function validity.

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Most popular questions from this chapter

Write the expression in algebraic form. \(\tan \left(\operatorname{arcsec} \frac{x}{3}\right)\)

Consider the function \(f(x)=\frac{4}{1+2^{4 / x}}\) (a) What is the domain of the function? (b) Use a graphing utility to graph the function. (c) Determine \(\lim _{x \rightarrow 0^{-}} f(x)\) and \(\lim _{x \rightarrow 0^{+}} f(x)\). (d) Use your knowledge of the exponential function to explain the behavior of \(f\) near \(x=0\).

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}+x-1, \quad[0,5], \quad f(c)=11 $$

In Exercises \(131-134,\) sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arcsin (x-1) $$

Rate of Change A 25 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate \(r\) of \(r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec}\) where \(x\) is the distance between the ladder base and the house. (a) Find \(r\) when \(x\) is 7 feet. (b) Find \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x \rightarrow 25^{-}\).
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