Question

Please explain in detail each step for me to understand

Answer 1

1. The given function is:

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]