Properties of Functions

3 2Sketch the following function 1f x x x x

0 0 1 intercept at 0,1x f y 3 20 0 1y x x x

20 0 1 1 1 1 1

1 or 1 cuts the axis at 1,0 , 1,0

y x x x x x

x x x

2' 3 2 1 0

3 1 1 0

1 at 13

f x x x

x x

SP x x

13

'' 6 2

'' 4 '' 0 hence maximum SP

'' 1 4 '' 0 hence minimum SP

f x x

f f x

f f x

3 2 321 1 1 1 13 3 3 3 3 27

3 2

321 max at ,27

1 1 1 1 1 0 min at 1,0

f

f

Odd FunctionsA function is said to be odd if

for every value in the

domain of The graph is symmetrical under 180 aboutthe origin

f x f x

x

y

y

3y x

even FunctionsA function is said to be even if

for every value in the

domain of The graph is symmetrical under in the axis

f x f x

xreflection

y

y

y

2y x

First derivative test.1. Differentiate2. Set derivative equal to zero3. Use nature table to determine the behaviour of the graph

Second derivative test1. Differentiate2. Set derivative equal to zero3. Find second derivative 4. Substitute x values in to second derivative5. If second derivative is positive, minimum6. If second derivative is negative, maximum7. If second derivative is zero or does not exist, use nature table

An asymptote is a line at which the rational

polynomial in the form of is undefinedf x

h xg x

1yx

1If 0, is undefined, since is not allowed0

x y

0is therefore a vertical asymptotex 10 0 which is also impossibleyx

0 is therefore a horizontal asymptotey

10 , is positivex yx

1, 0 from above is positivex yx

10 , is negativex y

x

1, 0 from below is negativex yx

2

3Sketch the graph of 2

xyx x

2

0 3 3 30 intercept is 0,2 20 0 2

x y y

2

30 0 3 0 intercept is 3,02

xy x xx x

To find any vertical asymptotes, we set the denominator equal to zero

2 2 02 1 0

2 and 1 are vertical asymptotes

x xx x

x x

1.110.9-1.9-2-2.1xy

Non vertical AsymptotesWhat happens to the y value if x tends to infinity

For the degree of the denominator is greater

than the numerator, hence the function tends to zero.

2

32

xyx x

0 is a non vertical asymptote.y

2

3Sketch the graph of 2

xyx x

2

0 3 3 30 intercept is 0,2 20 0 2

x y y

2

30 0 3 0 intercept is 3,02

xy x xx x

To find any vertical asymptotes, we set the denominator equal to zero

2 2 02 1 0

2 and 1 are vertical asymptotes

x xx x

x x

1.110.9-1.9-2-2.1xy

Non vertical AsymptotesWhat happens to the y value if x tends to infinity

For the degree of the denominator is greater

than the numerator, hence the function tends to zero.

2

32

xyx x

0 is a non vertical asymptote.y

2

A function is defined by

2 11

x xf xx

y f x

y f xFind the coordinates off the points wherethe graph crosses the coordinate axes

Find the equation of all vertical and non vertical asymptotes

Find the coordinates of any stationary points, and, if theyexist determine their nature.

y f xSketch the graph of

f x kState the range of values of the constant k such that the equation has no real solution

22 0 0 1 10 10 1 1

f

intercept at 0,1y

222 10 2 1 0

12 1 1 0

1 and 12

x x x xx

x x

x x

1intercept at ,0 and 1,02

x

Vertical Asymptote when1 0 1x x

2

2

2 31 2 1

2 23 13 3

2

xx x x

x xxx

22 31

y xx

2 3x y x 2 3 is a non vertical asymptotey x

0.9 1 1.1xy

x y

x y

122 3 2 3 2 11

y x x xx

22

22 2 1 21

dy xdx x

2

2

2

20 21

2 1 2 0

1 1

1 11 1 21 1 0

x

x

x

xxx

2

23

32

2 2 1

44 11

dy xdxd y xdx x

2

32

40 : 4 0,1 max0 1

d yxdx

2

32

42 : 4 2,9 min2 1

d yxdx

22 2 2 3 9 2,92 1

f SP

20 2 0 3 1 0,10 1

f SP