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.

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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

Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−1, 5) and is perpendicular to the line whose equation is x = 6.

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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

The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x+2)³

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