Functions 1

1. f(x) = 2x2 and g(x) = 5x – 4.

(a) Find f(g(2)).

(b) Find a formula for f(g(x)).

2. f(x) = 1 - 3x

2 and g(x) =

3 3x

2

+ x ≠ -1,0

(a) Given h(x) = f(g(x)), find a formula for h(x).

(b) State the connection between f(x) and g(x).

3. f(x) = 6x2 – 4x and g(x) = 2 x ,

6 -x 3

1≠

(a) Show that g(f(x)) = 1) -1)(x x 3(6

1

+.

(b) State a suitable domain for g(f(x)).

4. f(x) = 0,1 x ,x

2 - 1 g(x) and

x- 1

2≠=

(a) find f(g(x))

(b) State the connection between f and g.

5. f(x) = (x – 1)(x + 3) and g(x) = x2 + 3.

Show that f(g(x)) – g(g(x)) = 2x2

6. The functions f and g, defined on suitable domains, are given by

f(x) = 1 x g(x) and 4 - x

12

+= .

(a) Find an expression for h(x), where h(x) = f(g(x)).

Give your answer as a single fraction.

(b) State a suitable domain for h.

7. On a suitable set of real numbers, functions f and g are defined by

3 - x

1 g(x) and

3 x

1 f(x) =+

=

Find f(g(x)) in its simplest form.

8. A function f is defined on the set of real numbers by f(x) = 0 x ,x

x- 4 ≠

Find in its simplest form an expression for f(f(x)).

9. f(x) = 2,0- x , 2 - x

2 g(x) and

2 x

4≠=

+

Find f(g(x)) in its simplest form.

10. 0 x,x

12 -3x g(x) and

x

5 -x f(x) ≠==

(a) Show that f(g(x)) = 2) 2)(x -3(x

3) -4)(x (3x

+

+

(b) State a suitable domain for f(g(x)).

11. Two functions are defined as f(x) = x2 + 1 and g(x) = 2 – x

2 .

(a) Find an expression for f(f(x)).

(b) Find a similar expression for g(g(x)) and hence show that

f(f(x)) + g(g(x)) = 6x2.

12. f(x) = 2x + 1 and g(x) = x2 + k, where k is a constant.

(a) Find an expression for (i) g(f(x)) (ii) f(g(x)).

(b) Show that g(f(x)) – f(g(x)) = 0 simplifies to 2x2 + 4x – k = 0.

(c) Find the value of k for which 2x2 + 4x – k = 0 has equal roots.

Functions 2

1. f(x) = 2x – 5 and g(x) = x2 + 1. Find a formula for f(g(x)).

2. f(x) = x2 – 2. Find a formula for f(f(x)).

3. The functions f and g are defined on suitable domains with

f(x) = x

4 2x g(x) and

1 x

2 +=

+

Show that g(f(x) = 2x + 4.

4. 3 -4x g(x) and - x 1 3x

4 )x(f

31

=≠+

=

(a) Find a formula for f(g(x)).

(b) State a suitable domain for f((g(x)).

5. The functions f and g are defined on suitable domains with

1 x g(x) and 1 - x

1 )x(f

2+==

(a) h(x) = g(f(x)). Find an expression for h(x). Give your answer as a single fraction.

(b) State a suitable domain for h(x).

6. 1 x 1 -x

1 g(x) and 2- x

4 2x

1 )x(f ≠=≠

+=

(a) Find a formula for f(g(x)).

(b) State a suitable domain for f(g(x)).

7. The function f, defined on a suitable domain, is 1 -x

x )x(f =

(a) Find a formula for f(f(x)).

(b) What can you say about the function f?

8. f(x) = 2sin x and g(x) = ) x(2π

+

(a) Given k(x) = f(g(x)), find a formula for k(x).

(b) Solve the equation k(x) = -1, for 0 ≤ x ≤ 2π

9. f(x) = 2x – 1 and g(x) = x2 + x.

(a) Find a formula for g(f(x)) in its simplest form.

(b) h(x) = g(f(x)) – f(g(x)). Find a formula for h(x).

(c) Solve the equation h(x) = 7.

10. f(x) = x2 – 2 and g(x) = 2x + 1

(a) Find expressions for f(g(x)) and g(f(x)).

(b) There is only one value of x for which f(g(x)) = g(f(x)), find this value of x.

11. f(x) = 3x – 10 g(x) = 4 – 2x h(x) = x)- 2(61

(a) k(x) = f(g(x)). Find k(x).

(b) Find a formula for h(k(x)).

(c) What is the connection between h and k?

12. f(x) = 3x – 2 and g(x) = 3x + 2

(a) Find formulae for f(g(x)) and g(f(x)).

(b) Find the least value of the product f(g(x)) × g(f(x)).

13. f(x) = x2 + 1, x ≥ 0 and g(x) = x - 1 , x ≥ 1

(a) Sketch f(x) for the given domain.

(b) Find an expression for f(g(x)).

(c) Hence, or otherwise sketch g(x) for the given domain.

14. f(x) = 2x2 + 1, x ≥ 0 and g(x) = 1) - x(

21 , x ≥ 1.

(a) Sketch f(x) for the given domain.

(b) Find an expression for g(f(x)).

(c) Hence, or otherwise, sketch the graph of g(x) for the given domain.

Graphs of Functions

1. The graph of the function f(x) is shown.

On separate diagrams sketch the graphs

of

(i) f(x) + 3

(ii) –f(x)

2. The diagram shows the graph of y = f(x).

Sketch the graph of y = 3 – f(x).

3. Part of the graph of y = g(x) is shown.

On separate diagrams sketch the graphs

Of

(i) y = -2g(x)

(ii) y = g(x – 4)

x

y

(3,0)

(2,-4)

x

y

5

(3,2)

x

y

-6

(1,-7)

3

4. The graph of y = g(x) is shown.

Sketch the graph of y = -g(x – 2).

5. The graph of y = f(x) is shown.

On separate diagrams sketch the

graphs of

(i) y = - ½f(x) + 2

(ii) y = f(-x) – 2

6. The graph of y = h(x) is shown.

On separate diagrams sketch the

graphs of

(i) y = 2h(-x)

(ii) y = h(3x) – 1

x

y

(-4,-3)

(1,5)

7-2-5

x

y

(3,-8)

(-2,4)

x

y

-6

(-3,-6)

(6,12)

12

7. The diagram shows the graph of y = f(x).

Sketch the graph of y = f(½x) +3.

8. (a) f(x) = x2 + 4x + 7. Express f(x) in the form (x + a)

2 + b.

(b) Sketch the graph of y = f(x) showing its turning point and where it cuts the y-axis.

(c) Hence sketch the graph of y = -f(x) + 3.

9. (a) f(x) = x2 – 6x + 1. Express f(x) in the form (x – a)

2 – b.

(b) Sketch the graph of y = f(x).

(c) Hence sketch the graph of y = f(-x) –2.

10 . (a) f(x) = 2x2 + 8x – 3. Express f(x) in the form a(x + b)

2 + c

(b) Sketch the graph of y = f(x).

(c) Hence sketch the graph of y = -f(x – 2).

x

y

6-8

(-6,-10)

(4,-4)

Graphs of Functions

The following rules apply to all graphs, y = f(x), you may be asked to

draw, whether polynomial, trigonometric or logarithmic.

y = f(x) + a Moves a graph vertically: up if a > 0 down if a < 0

Example 1

2

Example 2

5

Example 3

4

y = f(x + a)

Moves a graph horizontally: left if a > 0 right if a < 0

Example 1

4

Example 2

5

Example 3

1500

y = – f(x)

Reflects a graph in the x – axis.

Example 1:

Example 2:

Example 3:

y = f(– x)

Reflects a graph in the y – axis.

Example 1:

Example 2:

Example 3:

y = kf(x)

Stretches or compresses a graph vertically.

Stretches if k > 1 Compresses if k < 1

Example 1:

Example 2:

y = ½f(x)

Example 3:

y = f(kx)

Stretches or compresses a graph horizontally.

Compresses if k > 1 Stretches if k < 1

Example 1:

Example 2:

y = f(½x)

Example 3:

Summary:

y = f(x) + a move vertically: up if a > 0, down if a < 0

y = f(x + a) move horizontally: left if a > 0 , right if a < 0

y = – f(x) reflect in x - axis

y = f(–x) reflect in y - axis

y = kf(x) stretch vertically if k > 1 , compress vertically if k < 1

y = f(kx) stretch horizontally if k < 1, compress horizontally if k > 1

NOTE: To draw something of the form y = 6 – f(x) it must be

done as y = –f(x) + 6 i.e. reflect in x – axis first then move

up by 6.

Example:

Functions from Graphs

1. The parabola opposite crosses the x-axis at (0,0) and (2,0)

and has a minimum turning point at (1,-6).

Find the equation of this parabola.

2. In the diagram A is the point (1,12).

Find the equation of f(x).

3. The cubic function shown has roots at

x = 0, x = 2 and x = 3.

It has a maximum turning point at (1,4).

Determine the equation of this cubic function

in the form

ax3 + bx

2 + cx + d

x

y

(2,0)O

(1,-6)

x

y

(3,0)O

A

y = f(x)

−1 1 2 3 4

−5

−4

−3

−2

−1

1

2

3

4

5

x

y

(1,4)

4. The function shown has zeros at x = 1, 4, and 6.

It has a maximum turning point at (5,8).

Find the equation of this cubic function.

5. The function opposite has roots of –1, 1 and 4 and it

crosses the y-axis at the point (0,16).

Find the equation of this function.

6. The parabola shown has a maximum turning point of

(4,32) and P is the point (8,0).

(a) Find the equation of the parabola.

(b) The line y = -2x + 16 intersects this parabola at

P and Q. Find the coordinates of Q.

7. A parabola passes through the points (0,0), (6,0) and (3,9)

as shown.

(a) The equation of this parabola can be written in the form

y = ax(b - x). Find the values of a and b.

(b) The line y = x + 4 intersects this curve at two points .

Find the coordinates of these points.

x

y

1 4 6

(5,8)

x

y

1 4-1

16

x

y

O P

(4,32)

Q

x

y

O

(3,9)

6

8. The diagram opposite is a sketch of the graph of a cubic

function y = f(x).

(a) If y = -16 is a tangent to the curve,

find a formula for f(x).

(b) The line y = 12x – 32 crosses this curve at

3 points.

Find the coordinates of these points.

9. The diagram shows a parabola passing

through (-1,0),

(0,p) and (p,0).

Show that the equation of the parabola can

be written as

y = p + (p – 1)x – x2.

x

y

6

-16

4

f(x)

x

y

-1 p

p

Functions from Graphs 2

1. The diagram shows a parabola with equation

y = ax(x – b).

Find the values of a and b.

2. The diagram shows a parabola with equation

y = kx(x – b).

Find the values of k and b.

3. The diagram shows a parabola with equation

y = ax(x – b).

Find the values of a and b.

4. The diagram shows a parabola with equation

y = kx(x + b).

Find the values of k and b.

5. The parabola opposite crosses the x-axis at (0,0) and (2,0)

and has a minimum turning point at (1,-6).

Find the equation of this parabola.

6. The parabola shown is of the form y = ax(x – b).

It has a maximum turning point of (4,32) and P

is the point (8,0).

(a) Find the equation of the parabola.

(b) The line y = -2x + 16 intersects this parabola at

P and Q. Find the coordinates of Q.

7. (a) Find the equation of the parabola, f(x),

shown opposite.

(b) Find the coordinates of P.

(c) Hence calculate the shaded area.

8. The diagram shows the graph of y = f(x).

The graph is of the form y = kx(x – b).

(a) Find a formula for f(x).

(b) Calculate the shaded area.

x

y

(2,0)O

(1,-6)

x

y

O P

(4,32)

Q

x

y P

1 5

-10

x

y

1 3 6

18

9. The diagram shows a parabola with equation

y = ax(x – b).

(a) Find the values of a and b.

(b) y = f /(x). Find a formula for f(x) given

f(3) = -4.

10. A parabola passes through the points (0,0), (6,0) and (3,9)

as shown.

(a) The equation of this parabola can be written in the form

y = ax(b - x). Find the values of a and b.

(b) The line y = x + 4 intersects this curve at two points .

Find the coordinates of these points.

11. The diagram shows a parabola with equation

y = k(x – a)(x – b).

(a) Find the values of k, a and b.

(b) y = f /(x). Find a formula for f(x) given

f(6) = -40.

12. In the diagram A is the point (1,12).

Find the equation of f(x).

x

y

O

(3,9)

6

x

y

(3,0)O

A

y = f(x)

13. The function shown has zeros at x = 1, 4, and 6.

It has a maximum turning point at (5,8).

Find the equation of this cubic function.

14. (a) The graph shown opposite crosses the x-axis

at (-1,0) and (2,0) and has a maximum turning

point of (0,4).

Find the equation of this graph.

(b) P is the point (3,14) and Q is (-1,-10).

Find the equation of the line PQ.

(c) The line PQ intersects the graph in (a)

at 3 points. If one of these points is

(4,20) find the other points of intersection.

15. The diagram opposite is a sketch of the graph of a cubic

function y = f(x).

(a) If y = -16 is a tangent to the curve,

find a formula for f(x).

(b) The line y = 12x – 32 crosses this curve at

3 points.

Find the coordinates of these points.

16. (a) The parabola opposite cuts the x-axis at

-1 and p and the y-axis at -2p.

Show that the parabola has equation

y = 2x2 +2x(1 – p) – 2p.

(b) The shaded area has a value

equal to 3

p10−.

Calculate the value of p.

x

y

1 4 6

(5,8)

x

y

4

-1 2

x

y

6

-16

4

f(x)

Polynomials

1. (a) Show that (x – 4) is a factor of x3 – 4x

2 – x + 4.

(b) Hence factorise x3 – 4x

2 – x + 4 fully.

2. (a) Show that (x + 1) is a factor of x3 – 7x – 6.

(b) Hence factorise x3 – 7x – 6 fully.

3. (a) f(x) = x3 + 3x

2 – 9x + 5. Factorise f(x) fully.

(b) Hence solve f(x) = 0.

4. f(x) = 4x3 – 8x

2 – x + 2. Given (x – 2) is a factor of f(x), solve f(x) = 0.

5. (a) f(x) = 2x3 + x

2 – 5x + 2. Show that (2x – 1) is a factor of f(x).

(b) Factorise f(x) completely.

6. (a) g(x) = 3x3 + 4x

2 – 5x – 2. Show that (3x + 1) is a factor of g(x).

(b) Hence solve g(x) = 0.

7. (a) Show that x = 1 is a solution of the equation x3 + x

2 – 10x + 8 = 0.

(b) Hence solve the equation x3 + x

2 – 10x + 8 = 0 completely.

8. Show that x = 2 is a solution to the equation 6x3 – 5x

2 – 17x + 6 = 0 and hence

solve the equation completely.

9. (a) Given that (x + 3) is a factor of x3 + 3x

2 – x + p, find the value of p.

(b) Hence solve the equation x3 + 3x

2 – x + p = 0 when p takes this value.

10. (a) Given (x + 1) is a factor of 2x3 – x

2 + kx – 9, find the value of k.

(b) Hence factorise fully 2x3 – x

2 + kx – 9 when k takes this value.

11. (a) f(x) = x3 – px

2 – 10x + 8p. Given (x – 2) is a factor of f(x), find p.

(b) Hence solve f(x) = 0 when p takes this value.

12. (x – 3) and (x + 3) are both factors of 2x3 – x

2 + px + q. Find p and q.

13. x = 1 and x = - 2 are both solutions to the equation 4x3 + x

2 + ax + b = 0. Find

the values of a and b.

14. (a) f(x) = 4x3 + 13x

2 + cx + d. Given (x – 1) and (x + 5) are both factors of f(x),

find c and d.

(b) Hence solve f(x) = 0 when c and d take these values.

15. (a) Show that (x – 1) is a factor of x3 + 5x

2 + 4x – 10.

(b) Hence, or otherwise, show that x = 1 is the only real solution to the equation

x3 + 5x

2 + 4x – 10 = 0.

16. Show that x = 4 is the only real solution to the equation 3x3 – 11x

2 – 16 = 0

17. A curve has equation y = 2x3 + 3x

2 – 6x – 4.

(a) Show that the line y = 3x + 6 intersects this

curve at the point (2,12).

(b) Find the other points of intersection of the

curve and the line y = 3x + 6.

18. A cubic has equation y = x3 – 3x

2 – 7x + 18 and

a parabola has equation y = x2 + 4x – 12.

(a) Show that the cubic and the parabola intersect

at the point (2,0)

(b) Find the other points of intersection of the

cubic and the parabola.

19. A curve has equation y = 3x3 + 6x

2 + 9x + 2.

(a) Show that the line y = 2x – 2 intersects the curve at the point (-1,-4).

(b) Show that there are no other points of intersection between the curve and the

line y = 2x – 2.

20. (a) Show that f(x) = 21 x

4 + 2x

2 + 24x – 1 has a stationary point when x = -2.

(b) Prove that f(x) has no other stationary points.

21. (a) y = 23 x

4 + 4x

3 + 6x

2 – 30x + 3. Show that ( )

231,1 − is a turning point on this

curve.

(b) Show that this curve has no other stationary points.

22. (a) Show that x = -1 is a solution to the equation x3 + px

2 + px + 1 = 0.

(b) Hence find the range of values of p for which all the roots of this equation

are real.

23. (a) Show that x = 2 is a solution to the equation 2x3 + kx

2 – 2kx – 16 = 0 .

(b) Hence find the range of values of k for which all the roots of this equation

are real.

Finding Roots of a Polynomial

1. f(x) = x3 – 7x + 6.

(a) Show that (x – 2) is a factor of f(x).

(b) Hence solve the equation f(x) = 0.

2. Show that 4 is a root of the equation x3 + 2x

2 – 15x – 36 = 0 and find the other

roots of this equation.

3. Show that 2 is a zero of 2x3 – 3x

2 – 3x + 2 = 0 and find the other zeros.

4. Find the roots of x3 + 6x

2 + 3x – 10 = 0.

5. y = x3 – 3x – 2 . Find the coordinates of the points where this curve cuts the x-axis.

6. Find the points where the curve y = x3 – 2x

2 – x + 2 cuts

(i) the y-axis (ii) the x-axis.

7. The line y = 4x + 10 and the curve y = x3 + 6x

2 + 3x – 20 intersect at 3 points.

One of these points is (-3,-2). Find the coordinates of the other points.

8. The tangent to the curve y = x3 – 7x + 6 at the point

(-1,12) has equation y + 4x = 8.

Find the coordinates of the other point of intersection

of the curve and this tangent.

9. The curve y = x3 – 10x + 6 and the line y = 2x – 10 intersect at two points.

Find the coordinates of these two points.

10. f(x) = x3 – 2x

2 – 5x + 6 and g(x) = x – 1.

(a) Show that f(g(x)) = x3 – 5x

2 + 2x + 8.

(b) Solve f(g(x)) = 0.

(c) State a suitable domain for f(g(x))

1.