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

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) = x³ +2

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Textbook Question

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

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

g(x) = f(x) - 1

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Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = -5, passing through (-4, -2)

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

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