MTH1010 - Functions and Their Applications Assignment

Assignment Task

Question

1. Are the following statements True or False? (brieáy justify your answer)

a) (i) θ  = {0}

(ii) x = {x}

(iii) θ = {θ}

(iv) θ =  {θ}

(b)The following sets have been expressed in set builder notation. Re-express them in list notation.

i) {ρ: ρ is a state capital, ρ is in Australiag}

(ii) {x : x = 2n 5 where n ² Ng}

(iii) {y : 2y 2 = 50 where y ² Zg}

(iv) {z : 3z = n ² where both n and z are natural numbers}

2. On the set A = f1; 2; 3; 4; 5g we may deÖne two functions f : A -> A and g : A -> A by

Fill in the Önal two rows by Önding the composite functions fog and gof

3. (a) If f and g are functions given

f : A -> R where f(x) = 1 /3 (1/ x+1)

g : B -> R where g(x) = 1/ 3x-1

Find

i) The implied maximal domains A and B for the functions f and g respectively.

(ii) The rule for fog and state itsírange.

(iii) The rule for gof and state itsírange. 

(b) (i) Find the function which is the inverse of

 (ii) State the domain and range of h ^-1 (x) from part i

(iii) Sketch the graphs of h (x) and h ^-1 (x) on the SAME set of axes.

4 (a)Let f : R -> R be the piece-wise function defined by

Find: (i) f (2) (ii) f (4) (iii) f (-1) iv f (-3)

(b) Sketch the graph of the piece-wise function f defined above.

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