Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.6

Chapter 1, Problem 1.6

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x²/⁵

Verified step by step guidance

To determine symmetry about the y-axis, check if replacing x with -x in the function yields the same function. For the given function y = x^(2/5), replace x with -x to get y = (-x)^(2/5).

Simplify the expression (-x)^(2/5). Since the exponent 2/5 is a positive rational number, (-x)^(2/5) simplifies to x^(2/5) because the square of a negative number is positive.

Since y = (-x)^(2/5) simplifies to y = x^(2/5), the function is symmetric about the y-axis.

To check for symmetry about the origin, replace both x with -x and y with -y in the original equation. This gives -y = (-x)^(2/5).

Since -y = x^(2/5) does not simplify to the original equation y = x^(2/5), the function is not symmetric about the origin. Therefore, the graph of the function is symmetric about the y-axis only.

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Key Concepts

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

Symmetry about the y-axis

A function is symmetric about the y-axis if replacing x with -x in the function yields the same output. Mathematically, this means that f(-x) = f(x) for all x in the domain of the function. This type of symmetry indicates that the graph of the function is a mirror image across the y-axis.

Symmetry about the origin

A function is symmetric about the origin if replacing x with -x and y with -y results in the same function. This is expressed as f(-x) = -f(x). Functions with this symmetry exhibit rotational symmetry of 180 degrees around the origin, meaning that if you rotate the graph, it looks the same.

Analyzing function symmetry

To determine the symmetry of a function, one can evaluate the function at both x and -x. By comparing the results, one can conclude whether the function is symmetric about the y-axis, the origin, or neither. This analysis is crucial for understanding the behavior of the graph and its visual representation.

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Properties of Functions

Related Practice

Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph two periods of the function f (x) = 3 cot (x/2) + 1.

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Textbook Question

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

f(x) = x⁻⁵

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Textbook Question

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = −√x Right 3

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Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph the upper branch of the hyperbola y² − 16x² = 1.

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Textbook Question

In Exercises 35 and 36, find the (a) domain and (b) range.

𝔂 = { -x - 2, -2 ≤ x ≤ - 1