Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 14

Chapter 3, Problem 14

In Exercises 11–26, determine whether each equation defines y as a function of x. x² + y = 25

Verified step by step guidance

Step 1: Recall the definition of a function. A relation defines y as a function of x if, for every value of x, there is exactly one corresponding value of y.

Step 2: Start with the given equation:

x^2 + y = 25

. Solve for y in terms of x by isolating y. Subtract

x^2

from both sides to get

y = 25 - x^2

Step 3: Analyze the resulting equation

y = 25 - x^2

. For any given value of x, there is only one corresponding value of y because the equation does not involve any operations (like a square root) that could produce multiple outputs for the same input.

Step 4: Conclude that the equation

y = 25 - x^2

defines y as a function of x because it passes the vertical line test. This means that for every x-value, there is exactly one y-value.

Step 5: Verify your conclusion by considering the graph of the equation. The graph of

y = 25 - x^2

is a parabola that opens downward, and it satisfies the condition of being a function.

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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 graph of a function. If any vertical line intersects the graph at more than one point, then the graph does not represent a function. This test helps in analyzing equations to see if they can be expressed as y in terms of x.

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. This involves isolating y on one side of the equation. If the resulting expression for y can be expressed as a single output for each input x, then y is a function of x.

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Related Practice

Textbook Question

Use the vertical line test to identify graphs in which y is a function of x.

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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) = 2x

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Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (0, -√2) and (√7,0)

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