/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 Draw the graphs of $$ f(x)=\... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 72

Draw the graphs of $$ f(x)=\sqrt{\sin ^{2} x} $$

Short Answer

Expert verified
The graph of the function \(f(x) = \sqrt{\sin^{2} x}\) will comprise of symmetrical 'waves' cresting at every integral multiple of \(\pi\) and descending back to 0 at multiples of \(2\pi\). It repeats itself every \(2\pi\) interval across all values of x and is always non-negative.

Step by step solution

01

Identify the Function

The given function is \(f(x) = \sqrt{\sin^{2} x}\). This is a real function as square roots of negative numbers are not defined in the real number domain.
02

Understand the Characteristics of the Function

Since \(f(x) = \sqrt{\sin^{2} x}\), we are dealing with the square function of sine and then taking the square root. Because square of any real number is always non-negative, the same logic applies here. Square of sine function sin^{2}x, will always be positive or zero, hence root over it will also be positive or zero. Another point to note is that sine is a periodic function with a period of \(2\pi\), so the graph of function will repeat itself every \(2\pi\) interval.
03

Graphing the Function

Because the function \(f(x) = \sqrt{\sin^{2} x}\) is always non-negative and repeats every \(2\pi\), the graph will look like a series of 'waves' that rise from 0 to 1 and back to 0 again. Each of these 'waves' is perfectly symmetrical, because the function is the same for \(x\) and \(-x\). Start by plotting a few key points on the x-axis such as \(0, \pi, 2\pi, 3\pi\) etc. For these x values, plot y values equal to \(f(x) = \sqrt{\sin^{2} x}\) which would be \(0, 1, 0, 1\)... respectively. There will be a 'wave' cresting at every multiple of \(\pi\) and hitting the x-axis at multiples of \(2\pi\). Draw the smooth curve to complete the graph.

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