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Please explain in detail each step for me to understand

Please explain in detail each step for me to understand-example-1

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1. The given function is:


g(x)=\begin{cases}5-x,-1\le x\le2 \\ x^2-1,2This is a piecewise function, the first part is given in the interval [-1,2], then let's solve this part for x=-1,0,1,2:<p>x=-1</p>[tex]g(-1)=5-(-1)=5+1=6

x=0


g(0)=5-0=5

x=1


g(1)=5-1=4

x=2


g(2)=5-2=3

Now, the second part is given in the interval (2,3]

Then let's solve for x=2.01 and x=3:

x=2.01


g(2.01)=(2.01)^2-1=4.0401-1=3.0401

x=3:


g(3)=(3)^2-1=9-1=8

Now, we can graph these ordered pairs:

This function is continuous on the interval [-1,3].

2. The given function is:


f(x)=\begin{cases}x+2,-1\le x<3 \\ 14-x^2,3\le x\le5\end{cases}

The first part of the function is given in the interval [-1,3).

Let's find f(x) for x=-1,0,1,2,2.99

x=-1


f(-1)=-1+2=1

x=0


f(0)=0+2=2

x=1


f(1)=1+2=3

x=2


f(2)=2+2=4

x=2.99


f(2.99)=2.99+2=4.99

The second part is given in the interval [3,5]

Then let's find f(x) for x=3,4,5

x=3


f(3)=14-3^2=14-9=5

x=4


f(4)=14-4^2=14-16=-2

x=5


f(5)=14-5^2=14-25=-11

Now, we can graph the function by graphing these ordered pairs:

The function is continuous in the interval [-1,5]

Please explain in detail each step for me to understand-example-1
Please explain in detail each step for me to understand-example-2
User Pieter Kuijpers
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