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, Problem 8T

(a)

To determine

To find: The value of f(2) and f(1) for the function f(x)={1xifx12x+1ifx>1

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)={1xifx12x+1ifx>1

Calculation:

The given function is piecewise defined function

So, for x1 function is defined as (1x) and for x>1 function is defined as (2x+1).

To calculate f(2) substitute x=2 in (1x) 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 (1x) because 1 is included in the domain of (1x).

f(1)=11=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)={1xifx12x+1ifx>1

Expert Solution

Explanation of Solution

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

For x1 the function is defined as (1x)

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

Sketch the graph of (1x) 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=1x.

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.

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