Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 19a

Chapter 3, Problem 19a

Determine whether each equation defines y as a function of x. y = √x +4

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).

Step 2: Analyze the given equation y = √x + 4. The square root function (√x) is defined only for x ≥ 0, as the square root of a negative number is not a real number.

Step 3: For each valid input x (x ≥ 0), the square root function produces exactly one output. Adding 4 to this output does not change the fact that there is only one output for each input.

Step 4: Since the equation y = √x + 4 produces exactly one value of y for each valid x, it satisfies the definition of a function.

Step 5: Conclude that the equation y = √x + 4 defines 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.

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 at more than one point, then the relation is not a function. This test helps to quickly assess whether an equation like y = √x + 4 defines y as a function of x.

Square Root Function

The square root function, represented as y = √x, is defined only for non-negative values of x, meaning x must be greater than or equal to zero. This function produces a unique output for each valid input, reinforcing the idea that it defines y as a function of x. Understanding the domain of this function is essential for analyzing the given equation.

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Related Practice

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.

Textbook Question

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

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Textbook Question

Use the graph of y = f(x) to graph each function g. g(x) = f(x+1)

Textbook Question

Determine whether each equation defines y as a function of x. $y = - \sqrt{x + 4}$

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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

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 (−2, 6) and is perpendicular to the line whose equation is x = -4.