Precalculus: Mathematics for Calculus - 6th Edition

6th Edition

ISBN: 9780840068071

Author: Stewart, James, Redlin, Lothar, Watson, Saleem

Publisher: Cengage Learning

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Question

Chapter 2, Problem 8T

(a)

To determine

To find: The value of f(−2) and f(1) for the function f(x)={1−x if x≤12x+1 if x>1

(a)

Expert Solution

Answer to Problem 8T

The value of f(−2) is 3 and f(1) is 0.

Explanation of Solution

Given:

The piecewise defined function is, f(x)={1−x if x≤12x+1 if x>1

Calculation:

The given function is piecewise defined function

So, for x≤1 function is defined as (1−x) and for x>1 function is defined as (2x+1).

To calculate f(−2) substitute x=−2 in (1−x) because −2 comes in domain of this function.

f(−2)=1−(−2)=1+2=3

Thus, the value of f(−2) is f(−2)=3.

To calculate f(1), substitute x=1 in (1−x) because 1 is included in the domain of (1−x).

f(1)=1−1=0

Thus, the value of f(−2) is 3 and f(1) is 0.

(b)

To determine

To sketch: The graph of function f(x)={1−x if x≤12x+1 if x>1

(b)

Expert Solution

Explanation of Solution

The behavior of piecewise defined function is different for different domains.

For x≤1 the function is defined as (1−x)

For x>1 the function is defined as (2x+1)

Sketch the graph of (1−x) and (2x+1) using transformation.

Sketch the graph of y=x, shift it horizontally right by 1 unit then takes its reflection in x-axis to get the graph of y=1−x.

Sketch the graph of y=x, stretch it vertically by factor of 2 and then shift it vertically up by 1 unit to get the graph of y=2x+1.

Sketch the piecewise defined function as shown in the figure below.

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 2, Problem 8T

From the above figure, it can be observed that there is a solid dot at (1,0) to show this point is included in the graph and there is open circle at (1,3) to show that this point is not included in the graph.

Chapter 2, Problem 7T

Chapter 2, Problem 9T

Chapter 2 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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Find a...Ch. 2 - Prob. 19P Ch. 2 - Minimizing a Sum Find two positive numbers whose...Ch. 2 - Prob. 21P Ch. 2 - Dividing a Pen A rancher with 750 ft of fencing...Ch. 2 - Fencing a Garden Plot A property owner wants to...Ch. 2 - Maximizing Area A wire 10 cm long is cut into two...Ch. 2 - Light from a Window A Norman window has the shape...Ch. 2 - Volume of a Box A box with an open top is to be...Ch. 2 - Area of a Box An open box with a square base is to...Ch. 2 - Inscribed Rectangle Find the dimensions that give...Ch. 2 - Minimizing Costs A rancher wants to build a...Ch. 2 - Minimizing Time A man stands at a point A on the...Ch. 2 - Bird Flight A bird is released from point A on an...Ch. 2 - Area of a Kite A kite frame is to be made from six...

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