Which conditions in the definition of continuity are not

Question

Question 1:

a) If possible, find the inverse of the function defined by y = x+2/ 2x-1

b) Do the following define one-to-one functions? Is it one-to-one or not?

i) y = cos x

ii) y = 3 -x²

iii) y = |x|

iv) y = sinhx

c) Functions f and g are defined by f (x) = x-2/5, g(x) = 5x +2,

i) Calculate f(g(x)).

ii) Calculate g(f(x)).

iii) Are f and g an inverse pair? Explain your answer.

d) Simplify cos(sin^-1 x).

e) Simplify ln(7x + 2) + ln(1/(7x+2)). What are the restrictions on x?

f) Use the definition of the hyperbolic sine function to evaluate sinh(0.76) , giving you answer correct to 2 decimal places.

g) Express cosh^-1 3 in terms of natural logarithms.

h) If possible, solve ln(x - 7) = ln(7 - x) where x is a real number.

i) Assume that f is a one-to-one function.

i) If f(2) = 6, what is f^-1 (6)?

ii) Evaluate f^-1 ( f (Π)).

iii) If b is real and f^-1 (b) = 3 , what is f (b)?

Question 2:

a) Use the definitions of cosh x and sinh x in terms of exponential functions to evaluate sinh x -cosh x.

b) Prove that tanh^-1 x = 1 ln(1+x/1-x).

c) Prove the change of base formula log_bx = log_ax/log_ab.

Question 3:

a) If h(x) = 4x + 2ln x, then what does h^-1 (4) equal?

b) The graphs of f and g are given. If possible, evaluate:

i) g( f (1)) =
ii) f (g(1)) =

iii) f^-1(-1) =

c) If possible, solve log_m (log_m x) = 1

Question 4:

a) Consider the complex number z = 5 + 12i . Evaluate:

i) Re(z)
ii) Im(z)
iii) |z|
iv) Arg (z) .

b) Express z = 1 + i√3 in polar form.

c) Evaluate e^3iΠ .

d) Solve z² = - i where z is a complex number. Express your answers in rectangular (or Cartesian) form.

e) Let

            4x -5,      x ≤ 4

f(x) =     3,           x> 4

Evaluate

i) lim  f(x)
x → 4^-
ii) lim  f(x)
x → 4+
iii) lim  f(x)
x → 4

f (x)
f (x)
f (x).

f) Evaluate

i)

lim_{x → 2 (}(3 - 2x)/ (3 + 2x))

ii) lim (4 - x²)/(x - 2)
x → 2

g) Evaluate

i)

lim  (x/(x-2))
x → 2-

ii)

lim ( x/x-2)
x → 2+

iii)

lim ( x/x-2 )
x → 2

h) Evaluate

lim (√x² +1)/(2x+3)
x → - ∞

i) Evaluate

lim (√(x² + x+ 1) -1)
x → ∞

Question 5:

a) Find the horizontal asymptote(s) of f(x) = e^-x/2e^x + e^-x.

b) A function y = f (x) is said to be continuous at x = c , if ALL three of the following statements are true:

i) f (c) exists

ii) lim f(x) exists
x → c

iii) lim f(x) = f(c).
x → c

Which conditions in the definition of continuity are not satisfied at x=1 following function?

                 x² -1/x-1       x ≠ 1
f (x) =

                 2,                 x = 1

c) Let z = c + id be a complex number, where c and d are real numbers. Prove that the product of z and its conjugate is a real number.

Question 6:

a) Evaluate the expression iⁱ , where i = √-1, in the form a + ib and a and b are realnumbers 

b) If possible, find all values of a that will make the following function continuous:

                 2x + 5        x < 1
f (x) =

                 3a,            x > 1

c) Explain the meaning of the mathematical concept "limit".