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Chapter 3: Q. 26 (page 171)

In Problems 25鈥�32, use the given functions f and g.

(a)f(x)=0(b)g(x)=0(c)f(x)=g(x)(d)f(x)>0

(e)g(x)鈮�0(f)f(x)>g(x)(g)f(x)鈮�1

f(x)=-x2+3g(x)=-3x+3

Short Answer

Expert verified

The required solution sets are

(a)x=-3,3

(b) x=1

(c)x=0,3

(d) x:-3<x<3

(e) x鈮�1

(f)x:0<x<3

(g)x:-2鈮�x鈮�2

Step by step solution

01

Part (a) Step 1. Given information

  • The given functions are f(x)=-x2+3g(x)=-3x+3.
  • The equation isf(x)=-x2+3g(x)=-3x+3.
02

Part (a) Step 2: Plot the graph and observe

  • Plot the graph of the function.

  • From the graph, it can be observed thatf(x)=0whenx=-3,3.
03

Part (b) Step 1. Given information

  • The given functions are f(x)=-x2+3g(x)=-3x+3
  • The equation isg(x)=0.
04

Part (b) Step 2. Plot the function and observe 

  • Plot the line in the graph obtained for the first function.

  • From the graph, it can be observed thatg(x)=0whenx=1.
05

Part (c) Step 1. Given information

  • The given functions are f(x)=-x2+3g(x)=-3x+3
  • The equation is f(x)=g(x).
06

Part (c) Step 2. Read the Graph 

  • For f(x)=g(x), the curves of both the functions must intersect.
  • From the graph, it can be observed that the functions intersect at (0,3)and(3,-6).
  • So, f(x)=g(x)at x=0,3.
07

Part (d) Step 1. Given information

  • The given functions are f(x)=-x2+3g(x)=-3x+3
  • The inequality is f(x)>0.
08

Part (d) Step 2.Find the region above the horizontal axis. 

  • The inequality holds when the curve of the function is above the horizontal axis.
  • According to the graph obtained in step 2 of part (b), the curve is above the horizontal axis when -3<x<3.
  • So, the solution set is x:-3<x<3.
09

Part (e) Step 1. Given information

  • The given functions are f(x)=-x2+3g(x)=-3x+3
  • The inequality isg(x)鈮�0.
10

Part (e) Step 2. Find the region on or below the horizontal axis. 

  • g(x)鈮�0when the line of the function is on or below the horizontal axis.
  • From the graph, the line is on or below the axis for x鈮�1.
11

Part (f) Step 1. Given information

  • The given functions are f(x)=-x2+3g(x)=-3x+3
  • The inequality isf(x)>g(x).
12

Part (f) Step 2. Read the graph

  • The inequality holds when the curve lies above the line on the graph.
  • From the graph, it can be observed that the curve is above the line when 0<x<3.
  • So, the solution set of the inequality is x:0<x<3.
13

Part (g) Step 1. Given information

  • The given functions are f(x)=-x2+3g(x)=-3x+3
  • The inequality isf(x)鈮�1.
14

Part (g) Step 2. Read the graph

  • The inequality holds when the curve lies above the value 1 on the vertical axis.
  • From the graph, the curve is above 1 when -2鈮�x鈮�2.
  • So, the solution set of the inequality is x:-2鈮�x鈮�2.

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