Continuity In a Function's Domain...4

[nycmathguy]

Calculus
Section 2.5

I say B is continuous because there are no holes, gaps and jumps in its domain.

 

[MathLover1]

correct

 

[nycmathguy]

correct

Questions

1. For problems 27-34, how is the domain found algebraically?

For 34, find domain algebraically .

2. For problems 27-34, how is the range found algebraically?

For 30, find range algebraically.

Thanks.

 

[MathLover1]

34.
g(t)=cos^-1(e^t-1)

domain: all real numbers except zero
cos^-1(e^t-1)=0.......apply inverse property
e^t-1=1
e^t=1+1
e^t=2.......take log
log(e^t)=log(2)
t*log(e)=log(2)
t*1=log(2)
t=log(2)
so, domain is
{t element R : t<=log(2)}


(-infinity,log(2)]



range:

Since cos^-1 is a decreasing function with range of 0<=cos^-1(t)< pi and e^t-1>-1, then
0<=cos^-1( e^t-1)< cos^-1(-1)
0<=cos^-1( e^t-1)< π
so, range is
{g element R : 0<=g<π}

For 30, follow steps above and find range algebraically.

 

[nycmathguy]

34.
g(t)=cos^-1(e^t-1)

domain: all real numbers except zero
cos^-1(e^t-1)=0.......apply inverse property
e^t-1=1
e^t=1+1
e^t=2.......take log
log(e^t)=log(2)
t*log(e)=log(2)
t*1=log(2)
t=log(2)
so, domain is
{t element R : t<=log(2)}


(-infinity,log(2)]



range:

Since cos^-1 is a decreasing function with range of 0<=cos^-1(t)< pi and e^t-1>-1, then
0<=cos^-1( e^t-1)< cos^-1(-1)
0<=cos^-1( e^t-1)< π
so, range is
{g element R : 0<=g<π}

For 30, follow steps above and find range algebraically.

Perfect. More notes for me to review when time allows.