1. The given function is:
[tex]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:
x=-1
[tex]g(-1)=5-(-1)=5+1=6[/tex]
x=0
[tex]g(0)=5-0=5[/tex]
x=1
[tex]g(1)=5-1=4[/tex]
x=2
[tex]g(2)=5-2=3[/tex]
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
[tex]g(2.01)=(2.01)^2-1=4.0401-1=3.0401[/tex]
x=3:
[tex]g(3)=(3)^2-1=9-1=8[/tex]
Now, we can graph these ordered pairs:
This function is continuous on the interval [-1,3].
2. The given function is:
[tex]f(x)=\begin{cases}x+2,-1\le x<3 \\ 14-x^2,3\le x\le5\end{cases}[/tex]
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
[tex]f(-1)=-1+2=1[/tex]
x=0
[tex]f(0)=0+2=2[/tex]
x=1
[tex]f(1)=1+2=3[/tex]
x=2
[tex]f(2)=2+2=4[/tex]
x=2.99
[tex]f(2.99)=2.99+2=4.99[/tex]
The second part is given in the interval [3,5]
Then let's find f(x) for x=3,4,5
x=3
[tex]f(3)=14-3^2=14-9=5[/tex]
x=4
[tex]f(4)=14-4^2=14-16=-2[/tex]
x=5
[tex]f(5)=14-5^2=14-25=-11[/tex]
Now, we can graph the function by graphing these ordered pairs:
The function is continuous in the interval [-1,5]
