BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2.2, Problem 84E
To determine

To describe whether the graph represent a function of x .

Expert Solution

Answer to Problem 84E

The equation, x=y2 does not define y as a function x because the equation gives two value of y for each value of x .

The equation, x=y3 does not define y as a function x because the equation gives two real value of y for each value of x .

For all rational value of n , the graph x=yn is the graph of a function of x .

Explanation of Solution

Given information :

The function is f(x)=xn for every integer n .

Concept used:

The graph of the function y=f(x) represent a function of x if and only if there is only one value of y for each value of x .

For the function, x=y2 ,

Solve for y in terms of x ,

  x=y2y=±x

The last equation gives two value of y that is +x or x for a given value of x .

Thus, the equation does not define y as a function x .

Similarly,

For the function, x=y3 ,

Solve for y in terms of x ,

  x=y3y=x3

The last equation gives two real valueand one complex value of y for a given value of x .

Thus, the equation does not define y as a function x .

If the function is in the form of y=xn , then only the equation define y as a function x .

Now,

The function in the form of x=yn will be the equation which define y as a function x if and only if n is a rational number.

Assume, n=12 , then the function is thus obtained is x=y12 ,

Solve for y in terms of x ,

  x=y12y=x2 [Square both sides]

The last equation gives only one real value of y for a given value of x .

Thus, the equation defines y as a function x .

Hence,

For all rational value of n , the graph x=yn is the graph of a function of x .

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