Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 6

Chapter 3, Problem 6

Determine whether each equation defines y as a function of x. 2x + y^2 = 6

Verified step by step guidance

Rewrite the given equation:

2x + y^2 = 6

. To determine if y is a function of x, we need to check if for every value of x, there is exactly one corresponding value of y.

Isolate the

y^2

term by subtracting

2x

from both sides:

y^2 = 6 - 2x

Take the square root of both sides to solve for y. Remember that taking the square root introduces both a positive and a negative solution:

y = \(\pm\)\(\sqrt{6 - 2x}\)

Analyze the result: The presence of the

\(\pm\)

symbol indicates that for a single value of x, there are two possible values of y (one positive and one negative).

Conclude that the equation does not define y as a function of x because it fails the vertical line test, meaning a single x-value corresponds to more than one y-value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we must check if for every x, there is a unique y. This means that for any given x, there should not be multiple y-values that satisfy the equation.

Vertical Line Test

The vertical line test is a visual method used to determine if a curve represents a function. If any vertical line drawn through the graph intersects it at more than one point, the relation is not a function. This test is particularly useful for analyzing graphs but can also inform our understanding of equations in implicit forms.

Solving for y

To analyze whether an equation defines y as a function of x, we often need to solve the equation for y. This involves isolating y on one side of the equation. If the resulting expression for y can yield multiple values for a single x, then y does not qualify as a function of x.

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Related Practice

Textbook Question

In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(3, −2), (5, −2), (7, 1), (4, 9)}

Textbook Question

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=5x-9 and g(x) = (x+5)/9

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Textbook Question

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-2, -6) and (3, −4)

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Textbook Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. (4, -1) and (3, −1)

Textbook Question

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (0, 0) and (3,-4)

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Textbook Question

Use the graph of y = f(x) to graph each function g.

g(x) = f(x-1) - 2

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