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Words 923

Pages 4

FUNCTION

Concept and Definition

1.

Determine the domain and range for each of the given function,

(a) f x x 1

Answer:

Since it is undefined for –ve values, it must be that x - 1 0 ie x 1

1

Domain ,

(b) f x

1 x 4

2

Answer: f (x) is undefined if x 2 4 0 , ie x 2 x 2 0 x = 2 or x = -2

The domain is all Real Numbers except 2 and -2.

domain , 2 atau 2, .

2

(c ) f x x 3

Answer:

All the real numbers ie ,

(d) f x x 5

Answer:

f x is defined for all real numbers

0 , x-50 x5 Domain 5,

1

MANUAL TUGASAN PELAJAR :MATEMATIK UNTUK EKONOMI DAN PENGURUSAN

x 1

(e) f ( x) x 2 6 x 5

Answer:

f (x ) is undefined when,

x 2 x 5x 0

( x 5)( x 1) 0 x 5

or x 1

Domain for f (x ) are all real numbers except –5 and –1

3

(f) f ( x)

2

4 x

jika jika x 1 x 1

Find the value of f ( 4 ) .

Answer:

f ( 4) 4 ( 4) 2 = 4 16 20.

(g) f x 2 x 3

Answer:

–2x + 3 > 0

–2x > –3

2x < 3 x < 3/2 = 1.5

Then the domain is "all x < 3/2".

2

MANUAL TUGASAN PELAJAR :MATEMATIK UNTUK EKONOMI DAN PENGURUSAN

2.

Given g ( x ) 3 x 2 x 5 , (i) find the domain and (ii) value of the function at g(z), g ( r 2 ) and g ( x h )

Answer:

Domain: All real numbers. g(z) = 3 z 2 z 5 g( r 2 ) = 3( r 2 ) 2 r 2 5 = 3r 4 r 2 5 g(x + h) = 3 ( x h) 2 ( x h) 5

= 3( x 2 2hx h 2 ) x h 5

= 3 x 2 6 hx 3h 2 x h 5

3. If f ( x ) x 2 , find

f ( x h) f ( x ) h Answer:

( x h)2 x 2 f ( x h) f ( x )

=

h h x 2 2hx h 2 x 2

=

h

=

2hx h2 h =

h( 2 x h)

= 2x + h h 4. Determine whether the given equation is a function

(i)

(ii)

2y + 3x = 6 y2 + 3x = 6

Answer:

2y + 3x = 6 is a function, because you can solve for y:

3

MANUAL TUGASAN PELAJAR :MATEMATIK UNTUK EKONOMI DAN PENGURUSAN

2y + 3x = 6

2y = –3x + 6 y = (–3/2)x + 3 y2 + 3x = 6 is not a function

Linear and Quadratic Functions

1. If a straight line passes through the point (1, a) and (4, -2) and is parallel to the line passing through the point (2, 8) and (-7, a + 4), what is the value of a?

Answer:

Since the two lines are parallel, m1 = m2

2a a4

3

9

-9 (-2 – a) = 3 (a – 4)

18 + 9a = 3a -12

6a = -30 a = -5

2. Given the equation y = 4x - 3, what is the corresponding change in y if

(a)

x increased by 1 unit?

(b) x decreases by 2 units?

Answer:(a)

If x increases 1 unit, y decreases 4 units.

(b)

If x decreases 2 units, y decreases 4 (-2) = -8 units.

4

MANUAL TUGASAN PELAJAR :MATEMATIK UNTUK EKONOMI DAN PENGURUSAN

3.

Given 2x + 3y = 4,

(a) Is the gradient of the equation positive or negative?

(b) When the value of x increases, does y increase or decrease?

(c) If x decreased by 2 units, what is the corresponding change in y?

Answer:

2x + 3y = 4

Deriving the slope-intercept form of the equation,

3y = 4 – 2x y= 4 2

x

3 3

2

, hence the gradient is negative.

3

(a)

m=

(b)

Since the gradient is negative, y decreases if x increases.

(c)

If x decreases by 2 units, y will increse by

2

2 4 unit.

3

3

4. (a) Determine the equation of the line that passes through the point (-2, 2) and parallel to the line 2x - 4y - 8 = 0.

(b) Find the equation of the line that passes through the point (2, 4) and perpendicular to the line 3x + 4y - 22 = 0.

Answer:

2x – 4y – 8 = 0

4y = 2x – 8 y= 1 x – 2.

2

Since the lines are parallel, m =

y–2=

1

.

2

1

[x – (-2)]

2

y=

1 x+3 2

5

MANUAL TUGASAN PELAJAR :MATEMATIK UNTUK EKONOMI DAN PENGURUSAN

(a)

3x + 4y – 22 = 0

4y = -3x + 22 y= 3 22

+

4

4

Since the lines are perpendicular, m =

the equation of the line is , y – 4 =

y=

4

.

3

4

(x – 2)

3

4

4

x+

3

3

5. Given the quadratic equation y = –x2 + 6x +7. Determines

(i)

(ii)

(iii)

if the parabola opens up or opens down. the coordinates of the vertex the x intercepts

Answer:

(i)

the parabola opens down because a < 0.

(ii)

the coordinates of the vertex

=

=

=

(

)

=

=3

(

)( )

(

)

= 16

the coordinates are (3, 16)

(iii)

6.

the x intercepts

Solve the simultaneous equations

5x−3y = 26 (1)

4x+ 2y = 34 (2) .

Solution

If y is eliminated, multiply equation (1) by 2 and equation (2) by 3. This gives

10x−6y = 52 (3)

12x+ 6y = 102 (4)

6

MANUAL TUGASAN PELAJAR :MATEMATIK UNTUK EKONOMI DAN PENGURUSAN

Therefore,

10x − 6y = 52

(3)

12x + 6y = 102

(4)

22x + 0y = 154

So x = 154/ 22 = 7.

By substituting,

5(7)−3y = 26

35−3y = 26

−3y = 26−35

−3y = −9 y= 3

Hence, x = 7, y = 3.

7…...

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