Functions Toolkit (DP IB Maths: AA HL)

The functions  and  are defined such that    and   .

Find , giving your answer in the form    where ,  and  are constants to be found.

Hence, or otherwise, find the coordinates of the vertex of the graph of  .

Find ,  giving your answer in the form    where ,  and  are constants to be found.

Hence, or otherwise, find the coordinates of the -intercept of the graph of  

Let    and   ,  where each function has the largest possible valid domain.

Write down the range of 

Write down the domain and range of .

Find

          (i)                                      

          (ii)      

Solve the equation .

The function  is defined by   ,  for  .

Write down the range of .

Write down an expression for .

Write down the domain and range of 

The perimeter, P, and area, A, of a given square can be expressed by  and  respectively, where  is the length of the side of the square.

Write down an expression for:

P in terms of A,

A in terms of P, .

Find the value of and .

The values of two functions,  and , for certain values of  are given in the following table:

		0	3
			8
	0		30

Find the value of .

Find the value of .

Given that  is a linear function, find .

Let    for .

Find .

Let  be a function such that  exists for all real numbers.

Given that , find .

Let the function  be defined by  ,  where  has its largest possible valid domain.

Find the domain and range of .

(i)     Find the value(s) of  for which   .

(ii)    Use your answer to part (b)(i) to explain why the inverse function  does not exist.

Let  and , both for .

Find

Find  in the form .

Solve the equation .

Express  in the form , where .

Given that  and , find .

Write   in the form  .

Explain why the function   defined by ,  does not have an inverse.

The function  defined by     has an inverse.

  (i)      Write down the smallest possible value of .

Given that  takes its smallest possible value:

   (ii)     Find the domain and range of .

   (iii)    Find the inverse function .

Solve .

Let  be an even function and let  be an odd function.  Both functions are defined for all real values of .

Prove the following statements:

is an odd function.

is an even function.

Determine whether or not it is possible for the function defined by 

to be even or odd, being sure to state clearly any conditions that apply.

The function  is defined by   ,  where , ,  and  are real constants with   .

Given that  is a self-inverse function, find the value of .

The functions  and  are defined such that    and   , both for  .

Find , giving your answer in the form .

Hence, or otherwise, find the -intercepts of the graph of .

Let .

Find the distance between the -intercept of the graph of    and the positive -intercept of the graph of  Your answer should be given as an exact value.

Let the function  be such that   .

Given that the inverse function  exists, and that the domain of  is as large as possible,

suggest a domain for and write down the corresponding range.

Based on your answer to part (a), find .

Let  .

Write down the coordinates of the -intercept of the graph of .

Given that  has the largest possible valid domain,

find the domain and range of .

Let the function  be defined by   ,  where  is a constant and where  has the largest possible valid domain.

Find the domain of .

Given that  that ,  find the value of .

Write down the equations of any vertical and/or horizontal asymptotes on the graph of .

The following diagram shows the graph of  ,  for a function  that has the domain .  Point A has coordinates   and point B has coordinates .  The -intercept of the function is  as shown.

 can be written as a piecewise function, where each of the two pieces is a linear function and where the domain of the first function is .

Write down  as a piecewise function.

Sketch the graph of   on the same grid above.

Consider the function  defined by   ,  .

Rewrite in the form  ,  where .

Given that   and that  ,  find .

The functions  and  are defined such that    and  ,  both for     .

Giving your answers in the form ,  find

Describe a single transformation that would map the graph of   onto the graph of  .

Given that ,  find the value of .

Let the functions  and  be defined by    and  ,  both for   .

Find

Find  in the form .

Solve the equation .

A rectangle has length  and width .

Find an expression for

the perimeter of the rectangle, P, in terms of .

the area of the rectangle, A, in terms of .

Show that .

The graph of the function P, for  , is shown below.

On the grid above, draw the graph of the inverse function 

Consider the function  defined by   ,  where  is the largest value such that  has an inverse.

  (i)     Find the value of .

  (ii)    On the same set of axes, sketch the graphs of  and .

  (iii)    Write down the domain and range of .

Find the inverse function .

Let the function  be defined by   .

  (i)     Solve .

  (ii)    Solve .

Consider the function   where   .

Show that:

if is even then .

if is odd then .

Consider the function  defined by     where  and  are real constants.

Find the possible values of   and  in the case where  is an even function.

Use proof by contradiction to show that   can never be an odd function.

Consider the function  defined by .

In the case where   is self-inverse, find the value of .

In the case when the graphs of and  intersect at exactly one point, find the possible values of .

A part of the graph of the function  is shown below.  

Explain why   does not have an inverse.

The domain of  is now restricted to     where     and   .  and are chosen so that  has an inverse and the interval  is as large as possible. 

Find the domain and range of