Functions - November 03, 2015

1a. [3 marks]

Consider a function f , defined by .

Find an expression for .

1b. [8 marks]

Let , where .

Use mathematical induction to show that for any

.

1c. [6 marks]

Show that is an expression for the inverse of .

1d. [6 marks]

(i) State .

(ii) Show that , given 0 < x < 1, .

(iii) For , let be the area of the region enclosed by the graph of , the x-axis and the line

x = 1. Find the area of the region enclosed by and in terms of .

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2a. [2 marks]

Express in the form where a, h, .

2b. [3 marks]

The graph of is transformed onto the graph of . Describe a sequence of

transformations that does this, making the order of transformations clear.

2c. [2 marks]

The function f is defined by .

Sketch the graph of .

2d. [2 marks]

Find the range of f.

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3a. [2 marks]

The function f is defined by , with domain .

Express in the form , where and .

3b. [2 marks]

Hence show that on D.

3c. [2 marks]

State the range of f.

3d. [6 marks]

(i) Find an expression for .

(ii) Sketch the graph of , showing the points of intersection with both axes.

3e. [7 marks]

(i) On a different diagram, sketch the graph of where .

(ii) Find all solutions of the equation .

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4. [18 marks]

Let .

(a) The graph of is drawn below.

(i) Find the value of .

(ii) Find the value of .

(iii) Sketch the graph of .

(b) (i) Sketch the graph of .

(ii) State the zeros of f.

(c) (i) Sketch the graph of .

(ii) State the zeros of .

(d) Given that we can denote as ,

(i) find the zeros of ;

(ii) find the zeros of ;

(iii) deduce the zeros of .

(e) The zeros of are .

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(i) State the relation between n and N;

(ii) Find, and simplify, an expression for in terms of n.

5a. [2 marks]

Sketch the graph of for .

5b. [3 marks]

Solve for .

6a. [2 marks]

The function f is defined by

Determine whether or not is continuous.

6b. [4 marks]

The graph of the function is obtained by applying the following transformations to the graph of :

a reflection in the –axis followed by a translation by the vector .

Find .

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7a. [2 marks]

Consider the following functions:

, ,

Sketch the graph of .

7b. [2 marks]

Find an expression for the composite function and state its domain.

7c. [7 marks]

Given that ,

(i) find in simplified form;

(ii) show that for .

7d. [3 marks]

Nigel states that is an odd function and Tom argues that is an even function.

(i) State who is correct and justify your answer.

(ii) Hence find the value of for .

8a. [4 marks]

The function f is defined as .

(i) Sketch the graph of , clearly indicating any asymptotes and axes intercepts.

(ii) Write down the equations of any asymptotes and the coordinates of any axes intercepts.

8b. [4 marks]

Find the inverse function , stating its domain.

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9a. [2 marks]

The diagram below shows a sketch of the graph of .

Sketch the graph of on the same axes.

9b. [1 mark]

State the range of .

9c. [4 marks]

Given that , find the value of and the value of .

10a. [4 marks] 7

The graphs of and are shown below.

Let f (x) = .

Draw the graph of y = f (x) on the blank grid below.

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11. [5 marks]

Let . The graph of f is transformed into the graph of the function g by a translation of

, followed by a reflection in the x-axis. Find an expression for , giving your answer as a

single logarithm.

12a. [3 marks]

The graph of is shown below, where A is a local maximum point and D is a local minimum

point.

On the axes below, sketch the graph of , clearly showing the coordinates of the images of the

points A, B and D, labelling them , , and respectively, and the equations of any vertical

asymptotes.

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13a. [2 marks]

Consider the following functions:

State the range of f and of g .

13b. [4 marks]

Find an expression for the composite function in the form , where .

13c. [4 marks]

(i) Find an expression for the inverse function .

(ii) State the domain and range of .

14a. [4 marks]

The quadratic function has a maximum value of 5 when x = 3.

Find the value of p and the value of q .

14b. [2 marks]

The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the

equation of the new graph.

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15a. [3 marks]

The diagram shows the graph of y = f(x) . The graph has a horizontal asymptote at y = 2 .

Sketch the graph of .

15b. [3 marks]

Sketch the graph of .

16. [6 marks]

A function is defined by . Find an expression for .

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17a. [4 marks]

Consider the equation .

Find the set of values of y for which this equation has real roots.

17b. [3 marks]

Hence determine the range of the function .

17c. [1 mark]

Explain why f has no inverse.

18a. [6 marks]

Given that ,

find , stating its domain;

18b. [1 mark]

find the value of x such that .

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19. [6 marks]

The graph below shows , where .

(a) On the graph below, sketch the curve .

(b) Find the coordinates of the point of intersection of the graph of and the graph of

.

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20a. [2 marks]

Consider the functions given below.

(i) Find and write down the domain of the function.

(ii) Find and write down the domain of the function.

20b. [4 marks]

Find the coordinates of the point where the graph of and the graph of

intersect.

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21a. [2 marks]

The diagram below shows the graph of the function , defined for all ,where

.

Consider the function .

Find the largest possible domain of the function .

21b. [6 marks]

On the axes below, sketch the graph of . On the graph, indicate any asymptotes and local

maxima or minima, and write down their equations and coordinates.

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22a. [5 marks]

Consider the function , .

Sketch the graph of , indicating clearly the asymptote, x-intercept and the local maximum.

22b. [6 marks]

Now consider the functions and , where .

(i) Sketch the graph of .

(ii) Write down the range of .

(iii) Find the values of such that .

23. [8 marks]

The functions f and g are defined as:

(a) Find .

(b) State the domain of .

(c) Find .

24. [6 marks]

Let .

If , find

(a) h(x) ;

(b) , where is the inverse of h.

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25. [5 marks]

The graph of is shown.

On the set of axes provided, sketch the graph of , clearly showing any asymptotes and

indicating the coordinates of any local maxima or minima.

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26. [7 marks]

(a) Sketch the curve , showing clearly the coordinates of the

points of intersection with the x-axis and the coordinates of any local maxima and minima.

(b) Find the values of x for which , .

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27. [16 marks]

A function is defined as , with and .

(a) Sketch the graph of .

(b) Show that f is a one-to-one function.

(c) Find the inverse function, and state its domain.

(d) If the graphs of and intersect at the point (4, 4) find the value of k .

28. [6 marks]

Consider the function f , where .

(a) Find the domain of f .

(b) Find .

30. [6 marks]

The function f is of the form , . Given that the graph of f has asymptotes x = −4

and y = −2 , and that the point lies on the graph, find the values of a , b and c .

31. [4 marks] 22

Shown below are the graphs of and .

If , find all possible values of x.

32. [8 marks]

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The graph of is drawn below.

(a) Find the value of a, the value of b and the value of c.

(b) Using the values of a, b and c found in part (a), sketch the graph of on the axes below,

showing clearly all intercepts and asymptotes.

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33. [6 marks]

(a) Express the quadratic in the form , where a, b, c .

(b) Describe a sequence of transformations that transforms the graph of to the graph of

.

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34. [6 marks]

A function f is defined by .

(a) Find an expression for .

(b) Solve the equation .

35. [8 marks]

Consider the function .

(a) Find the largest possible domain of f.

(b) Determine an expression for the inverse function, , and write down its domain.

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36. [5 marks]

The diagram shows the graphs of a linear function f and a quadratic function g.

On the same axes sketch the graph of . Indicate clearly where the x-intercept and the asymptotes

occur.

37. [8 marks]

Consider the function , where .

(a) Given that the domain of is , find the least value of such that has an inverse function.

(b) On the same set of axes, sketch

(i) the graph of for this value of ;

(ii) the corresponding inverse, .

(c) Find an expression for .

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38a. [2 marks]

The graph of is transformed into the graph of .

Describe two transformations that are required to do this.

38b. [4 marks]

Solve , .

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