Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 8

Chapter 3, Problem 8

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=√x? What is its domain?

Verified step by step guidance

Recall the definition of the function ƒ(x) = $\sqrt{x}$, which represents the principal (non-negative) square root of x.

Understand that the graph of ƒ(x) = $\sqrt{x}$ starts at the point (0,0) because $\sqrt{0}$ = 0, and it only includes values where x is greater than or equal to zero, since the square root of a negative number is not a real number.

Identify the graph that begins at the origin (0,0) and increases gradually to the right, forming a curve that rises slowly as x increases; this is the characteristic shape of the square root function.

Determine the domain of ƒ(x) = $\sqrt{x}$ by considering all x-values for which the function is defined. Since the square root requires non-negative inputs, the domain is all real numbers x such that x $\geq$ 0.

Express the domain in interval notation as [0, $\infty$), indicating that the function includes zero and all positive real numbers.

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Key Concepts

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

Square Root Function

The square root function, denoted as ƒ(x) = √x, outputs the non-negative value whose square is x. It is defined only for x ≥ 0 because the square root of a negative number is not a real number. Understanding this function's shape helps in identifying its graph.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For ƒ(x) = √x, the domain includes all real numbers x ≥ 0, since the square root of negative numbers is not real. Recognizing the domain is essential for graph interpretation.

Graphing Basic Functions

Graphing basic functions involves plotting points that satisfy the function's equation and understanding their general shape. The graph of ƒ(x) = √x starts at the origin (0,0) and increases slowly to the right, forming a curve in the first quadrant. Familiarity with these shapes aids in matching functions to their graphs.

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Related Practice

Textbook Question

Determine whether each statement is true or false. If false, explain why. The graph of y = x² + 2 has no x-intercepts.

Textbook Question

Fill in the blank(s) to correctly complete each sentence. The function $g (x) = √ x$ has domain ________.

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Textbook Question

Let ƒ(x)=x²+3 and g(x)=-2x+6. Find each of the following. (ƒ+g)(3)

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Textbook Question

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=∛x? Is there any open interval over which the function is decreasing?

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Textbook Question

Without actually graphing, identify the type of graph that each equation has.

$x^2+y^2=144$

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Textbook Question

Determine whether each statement is true or false. If false, explain why. The midpoint of the segment joining (0, 0) and (4, 4) is 2.