Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 18a

Chapter 3, Problem 18a

In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²

Verified step by step guidance

Step 1: Recall the definition of a function. A function is a relation where each input (x) corresponds to exactly one output (y). To determine if the given equation defines y as a function of x, we need to check if each x-value produces a unique y-value.

Step 2: Start with the given equation:

4x = y^2

. Rearrange it to isolate y. Divide both sides of the equation by 4 to get

x = rac{y^2}{4}

Step 3: Solve for y by taking the square root of both sides. Remember that taking the square root introduces both a positive and a negative solution. This gives

y = \(\text{±}\)\(\text{√}\)(4x)

Step 4: Analyze the result. Since y can take two values (positive and negative) for a single x-value, the equation does not define y as a function of x. A function must have only one output for each input.

Step 5: Conclude that the equation

4x = y^2

does not define y as a function of x because it fails the vertical line test, which states that a vertical line drawn through any x-value should intersect the graph at most once.

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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 between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines y as a function of x.

Vertical Line Test

The vertical line test is a visual way to determine if a curve is a function. If any vertical line intersects the graph of the relation more than once, then the relation is not a function. This test helps to quickly assess whether an equation can be expressed as y in terms of x without ambiguity.

Solving for y

To determine if an equation defines y as a function of x, it is often necessary to solve the equation for y. In the case of the equation 4x = y², rearranging it to express y in terms of x reveals whether y can take on multiple values for a single x. This step is essential for understanding the relationship between the variables.

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Related Practice

Textbook Question

Use the graph to determine (a) the function's domain, (b) the function's range, (c) the x-intercepts, if any, (d) the y-intercept, if there is one, (e) intervals on which the function is increasing, decreasing or constant, (f) the missing function values, indicated by question marks, below each graph.

Textbook Question

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

g(x) = f(x-1)

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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. (-1/4, -1/7) and (3/4, 6/7)

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Textbook Question

Find the domain of each function. f(x) = √(x - 3)

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Textbook Question

Find the midpoint of each line segment with the given endpoints. (6, 8) and (2, 4)

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Textbook Question

Find the domain of each function. f(x) = 1/√(x - 3)

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