Functions Toolkit (DP IB Maths: AI HL)

The functions f and g are defined such that   and .

Find ,  giving your answer in the form    where m, h and k 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 y-intercept of the graph of

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

Write down the range of f.

Write down the domain and range of g.

Find

Solve the equation

The function f is defined by ,  for .

Write down the range of f.

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 x is the length of the side of the square.

Write down an expression for: 

Find the value of k and .

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

Find the value of f ^-18.

Find the value of

Given that f (x) is a linear function, find f (x).

Let ,  for .

Find f ^-1(2).

Let g be a function such that g^-1 exists for all real numbers.

Given that g(14) = 3, find .

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

Find the domain and range of f.

Let and, both for .

Find

Find in the form .

Solve the equation

Given that g and , find a possible expression for .

The function g defined by g has an inverse.

Write down the smallest possible value of .

Given that  takes its smallest possible value:

Solve = 21.

The function is defined by .

Find .

It is always true that the graphs of a function and its inverse will be reflections of each other in the line .

Based on the answer to part (i), state what else will be true about the graphs of  and .

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

Find, giving your answer in the form  .

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

Let .

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

Let the function f  be such that  .

It is given that the inverse function f ^-1 exists, and that the domain of f  is as large as possible,

suggest two possible domains for f and write down the corresponding ranges.

Find what the value of would be for each of the domains suggested in part (a).

Let.

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

Given that  f  has the largest possible valid domain,

find the domain and range of  f .

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

Find the domain of f.

Given that as gets large tends towards the value −7, 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 f that has the domain.  Point A has coordinates (-3, 2.5) and point B has coordinates (3,-2.5).  The x-intercept of the function is (2, 0) as shown.

f 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 .

Show that .

Hence show that .

Given that  and that , find a possible expression for g.

The functions f and g 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 p.

Let the functions f and g 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.

Find the value of .

On the same set of axes, sketch the graphs of and .

Write down the domain and range of .

Find the inverse function .

Let the function g be defined by g.

Solve 

Solve 

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