Inequalities (DP IB Maths: AA HL)

Consider the function defined by .

(i)

Sketch the graph of .

Consider the function

Fully factorise .

Find the values of  such that the equation  has real solutions and the equation  has no real solutions.

Find the set of values of for which 

          (i)     

          (ii)    

Consider the functions

and  

where  is real constant such that  

(a)  In terms of the constant , find 

          (i)     the values of  for which  and  are undefined, 

          (ii)    the -coordinate of any intersections between the graphs of and .

In the case , find the set of values of  in terms of  for which 

          (i)     , 

          (ii)    .

Consider the functions defined by  and  where  and  are positive constants. Show that

 for  .   .

Consider the functions defined by and where . Given that  only for find the values of a and b.

The function defined by  can be factorised into the form  where  and  are positive integers such that .

The function  is such that 

Find a polynomial, of the lowest degree possible, that satisfies the condition .

The region R is defined by the three straight lines given by the inequalities

The function  is defined by . Find the largest domain of  such that the graph of lies within the region R. Give answers as exact values where appropriate.

Consider the functions defined by  and All three functions have the domain  

On the same diagram, sketch the graphs of and .

Find the exact values for  such that