Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 17

Chapter 3, Problem 17

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

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Recall the definition of a function: for each input value of \( x \), there must be exactly one output value of \( y \).

Given the equation \( y^2 = x \), solve for \( y \) in terms of \( x \) by taking the square root of both sides: \( y = \pm \sqrt{x} \).

Notice that for each positive value of \( x \), there are two possible values of \( y \) (one positive and one negative), which means \( y \) is not uniquely determined by \( x \).

For \( x = 0 \), there is exactly one value of \( y \) (which is 0), but for other values of \( x \geq 0 \), there are two values of \( y \).

Since the equation does not assign exactly one \( y \) for each \( x \), it does not define \( y \) as a function of \( x \).

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Key Concepts

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

Definition of a Function

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 check if for every x there is only one y. If any x maps to multiple y-values, the relation is not a function.

Solving Equations for y

To analyze if y is a function of x, we often solve the equation for y explicitly. For y² = x, solving for y gives y = ±√x, indicating two possible y-values for each positive x. This suggests the relation may not define y as a function of x.

Vertical Line Test

The vertical line test is a graphical method to determine if a relation is a function. If any vertical line intersects the graph more than once, the relation is not a function. For y² = x, the graph is a sideways parabola, failing the vertical line test.

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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

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

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 average rate of change of the function from x₁ to x₂. f(x) = √x from x₁ = 4 to x₂ = 9

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. (7/3, 1/5) and (1/3, 6/5)

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