   Chapter 3.R, Problem 20E

Chapter
Section
Textbook Problem

# 17-28 Use the guidelines of Section 3.5 to sketch the curve. y = x 1 − x 2

To determine

To sketch:

The graph of the function y=x1-x2

Explanation

1) Concept:

i) A domain is the set of xvalues that satisfy the function.

ii) To find x-intercept, put y=0,and to find y-intercept, put x=0in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behaviour of function. Thus, if f-x=fx, then it is an even function, so it has y-axis symmetry. If f-x=-fx, then it is an odd function, so it has x-axis symmetry. And if f-x-fxfx, then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical, and slant asymptote, follow the rules.

v) A function is increasing if f'x>0 and decreasing if f'x<0 in that particular interval.

vi) The number f(c)is a local maximum value of f if fcf(x)when x is near c and is a local minimum value of fif fc f(x)when x is near c.

vii) If f''x>0, the function is concave up and if f''x<0, the function is concave down in that particular interval. And if f''x=0, give the values of inflection points

2) Given:

The function y=x1-x2

3) Calculation:

A) Domain

The given function is y=x1-x2

The function in defined except the zeros of the denominator

Therefore 1-x2=0 becomes x= ±1

Domain of the function is (-,-1)(-1 ,1)(1, )

B) Intercepts

For y intercept, plug x=0  in the given function and solve it

y=01-02=0

Therefore, y- intercept of function is (0,0)

Now for x-intercept set y= 0 then equation becomes

x1-x2=0

We get, x = 0

Therefore x-intercepts is (0,0)

C) Symmetry

For symmetry replace x by (-x), therefore,

f-x=-x1--x2= -x1-x2= -f(x)

Since, f-x=-f(x)

f  is an odd function

Therefore, the graph is symmetric about the origin

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