Assignment 1 Complete this assignment after you have finished Unit 3, and submit your work to your tutor for grading. Remember to attach a tutor-marked exercise form (paper or electronic). Be certain...

1. Indicate whether each of the following functions is invertible in the given interval. Explain
a. [ )
The given function is a continuously decreasing function. Any line drawn parallel to x – axis
cuts the graph of the function at a single point. Hence we can say that the given function is
invertible.

. ( ) ( ]
The given function is a oscillating. Any line drawn parallel to x – axis cuts the graph of the
function at more than a single point. Hence we can say that the given function is not
invertible
.
c.

( ]
The given function is decreases and increases. Lines drawn parallel to x – axis (near y=1) cuts
the graph of the function at more than a single point. Hence we can say that the given
function is not invertible.

2.
a. Use Definition 1 on page 38 of the textbook to show that the function is one – one
( )



Sol:
Let x1, x2 be such that,
( ) ( )







(

)



(

)



(

)


(

)

Hence the given function is one –one.
. Find the inverse function, f-1(x)
Sol:
Let the f-1(x) =y
( )







(
( )
)








( )



c. Give the domain and range of the functions f and f-1.
Sol:
The functions f and f-1 cannot take the values x=5/2 and x=-3/2 respectively as these
values make the denominators 0, which in turn makes the value of the function undefined.
Hence the domain of the function f is ,


- and the domain of the function f-1 is ,

-
The range of function f cannot have the value that is not in the domain of the function f-1 and
vice – versa.
Hence the range of the function f is ,


- and the range of the function f-1 is ,


-
3. Find the solution of the following equations, if the solution exists.
a.




Sol:
The equation can be reduced to,


Taking logarithm on both sides and solving the quadratic equation obtained, we get,
. ( ) ( )
Sol:
Using the property of the logarithm that ln(a)+ln(b)=ln(ab), we can write,
(( )( )) ( )
( )
Solving the quadratic equation we get the roots to be,

√


c. ( )
Sol:
Taking cos on both sides, we get,




d. ( )
Sol:
Taking sec on both sides, we get,
( )

( )


4. Use logarithmic differentiation to find,
a.


Sol:
Let


( )
( ( )) ( ) ( ) ( )

( )
( )
( ) (√ )
( ) (√ )




( )

( ) (√ )
( ) (√ )

( ) ( )
( )
.



Sol:
Let ( )
We know that, ( √ ) ( )
Differentiating this, we get,

√


In this case, y=e2x.
Hence,
( )


√









( )
5. Evaluate the following limits, justifying your answers. If a limit does not exist, explain why.
a.



This function is taking the form


. No matter the values of x, the value of cosx and sinx
emain bounded in [-1, 1].
Hence we can say that, both cosx ,sinx divided by x, as x approaches infinity, approach 0.
Divide numerator and denominator with x3.









.
(


)


Assuming , this can be...