Advanced Placement Calculus Graphical Numerical Algebraic Sixth Edition High School Binding Copyright 2020

6th Edition

ISBN: 9781418300203

Author: Prentice Hall

Publisher: Prentice Hall

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Question

Chapter 0.5, Problem 51E

(a)

To determine

To find: The convincing argument that $f$ is a one to one function.

(a)

Expert Solution

Answer to Problem 51E

$fx1=fx2$ and $x1=x2$ therefore by definition this function is one-to-one function.

Explanation of Solution

Given information: The function is: $y=f(x)=mx+b$

Calculation:

$fx1=mx1+b fx2=mx2+b$

Therefore those two functions will be equal when:

$mx1+b=mx2+bmx1=mx2x1=x2$

Keep in mind that we may divide by $m$ in the final step because $m≠0$ .

Therefore the required $fx1=fx2$ and $x1=x2$ therefore by definition this function is one-to-one function.

(b)

To determine

To find: The for the inverse of $f$ .and explain the slope $f−1$ related.

(b)

Expert Solution

Answer to Problem 51E

$f−1(x)=xm−bm$

The slopes are inverse.

Explanation of Solution

Given information: The function is: $y=f(x)=mx+b$

Calculation:

Switch $x$ and $y$ to solve for $y$ :

$x=my+bmy=x−by=x−bmy=xm−bm$

The inverse function of $f$ is:

$f−1(x)=xm−bm$

The slope of $f−1$ is $1m$ .

(c)

To determine

To explain: The graph of the function is inverse.

(c)

Expert Solution

Answer to Problem 51E

They are a pair of parallel lines as well.

Explanation of Solution

Given information: The function is: $y=f(x)=mx+b$

Explain:

Let f and g represent linear functions whose graphs are parallel lines, and thus they have the same slope $m≠0$ .

Then their respective inverses $f−1$ and $g−1$ will have the same slope $1m$ . In other words, the graphs of their inverses will be a pair of parallel lines as well.

(d)

To determine

To explain: The graph of the function is inverse.

(d)

Expert Solution

Answer to Problem 51E

They are a pair of perpendicular lines as well.

Explanation of Solution

Given information: The function is: $y=f(x)=mx+b$

Explain:

From 51(b):

$f−1 as 1m$ ,the slope $f−1$ as $1m$ and $g−1$ as $−1m−1=−m$

Since,

$1m(−m)=−1$

The graph of $f−1$ and $g−1$ are perpendicular as well.

Chapter 0.5, Problem 50E

Chapter 0.5, Problem 52E

Chapter 0 Solutions

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Knowledge Booster

The convincing argument that f is a one to one function.

Solution Summary: The author explains the convincing argument that f is a one-to-one function.

To find: The convincing argument that $f$ is a one to one function.

Answer to Problem 51E

Explanation of Solution

To find: The for the inverse of $f$ .and explain the slope $f−1$ related.

Answer to Problem 51E

Explanation of Solution

To explain: The graph of the function is inverse.

Answer to Problem 51E

Explanation of Solution

To explain: The graph of the function is inverse.

Answer to Problem 51E

Explanation of Solution

Chapter 0 Solutions

The convincing argument that f is a one to one function.

Solution Summary: The author explains the convincing argument that f is a one-to-one function.

To find: The convincing argument that f<m:math><m:mi>f</m:mi></m:math> is a one to one function.

Answer to Problem 51E

Explanation of Solution

To find: The for the inverse of f<m:math><m:mi>f</m:mi></m:math> .and explain the slope f−1<m:math><m:mrow><m:msup><m:mi>f</m:mi><m:mrow><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow></m:math> related.

Answer to Problem 51E

Explanation of Solution

To explain: The graph of the function is inverse.

Answer to Problem 51E

Explanation of Solution

To explain: The graph of the function is inverse.

Answer to Problem 51E

Explanation of Solution

Chapter 0 Solutions

To find: The convincing argument that $f$ is a one to one function.

To find: The for the inverse of $f$ .and explain the slope $f−1$ related.