Maths Worksheet - Functions, Inverses And Logarithms

SAEP Matric Success – Zisukhanyo Maths Worksheet Tuesday 24th February 2009 Functions, Inverses and Logarithms Definition of a function. 1. State, with explanation, which of the following functions are one-to-one: a) f ( x ) = 3 − x b) g ( x ) = x 3 c) h( x ) = x d) p( x ) = −3 1 x f) k ( x ) = x 2 − 2 x − 8

e) t ( x ) =

Linear 2. a) if the point (p;7) lies on the line y = 3 x − 5 what does p equal? b) What are the co-ordinates of the points of intersection of the graph

1 1 x − y = 1 with 2 4

the axes? c) Which one of the following lines is parallel to 2 x + y −1 = 0 ? y = 2x +3 ; 2 y + x +1 = 0 ; 2x + y = 3 ; x = 2 y +1 d) Which one of the following lines is perpendicular to 2 x + y −1 = 0 ? y = 2x +3 ; 2 y + x +1 = 0 ; x = 2 y +1 ; 2 y + x =1 Quadratics 3. Solve for x a) 12 x 3 − 4 x 2 − 5 x = 0 b) 3(1 − x ) 2 − 2(1 − x ) −1 = 0 3x 2 + 2 x + 3 12 c) + 2 =7 1 3x + 2 x + 3 3x 2 + x −1 1 d) + 2 =0 1 3x + x − 3 e) 14 +17 x − 2 x 2 = x + 2 f) 6 − x + 2 = x + 2 2 g) ( 2 x 2 + x ) − 6 x 2 − 3x + 2 = 0 h)

3 x −1 + 1 =

6 3 x −1

1  x ≠  3 

4. Solve for x by completing the square 2ax 2 − 5bx + ab = 0

5. a) Solve for x: x − 2 = 2 − x b) Hence find the value(s) of p if

p 2 − p −2 = 2 + p − p 2

6. a) Sketch the graph of y = −x 2 − 3x + 4 b) On the same set of axes, draw the graph of x + y = 4 Inverses 7. For each of the functions listed below, restrict the domain if necessary then find an inverse function f −1 , give the domain and range of f −1 , and sketch the graphs y = f ( x ) , y = x and y = f −1 ( x ) on the same system of axes [ x ∈ R ] : a) f ( x ) = 3 x b) f ( x ) = −x c) f ( x ) = x − 3 d) f ( x ) =

1 x −1 2

→ − x 8. g −1 : x  a) Write down the domain and range of g −1 b) Determine g. c) Draw sketch graphs of g and g −1 on the same system of axes.

Remainder Theorem 9. Use the Remainder Theorem to determine the remainder when a(x) is divided by b(x) in each of the following cases: b( x ) = x + 2 a) a( x ) = 2 x 3 − x 2 + 3 x − 8 2 b( x ) = x +1 b) a( x ) = 5 x − 6 x + 2 3 b( x ) = x − 2 c) a( x ) = x − 2 x −12 4 2 b( x ) = 2 x +1 d) a( x ) = 8 x − 4 x − 5 10. When f ( x ) = ax 3 + bx 2 − 4 x + 6 is divide by x − 2 , the remainder is 2. When it is divided by x −1 , the remainder is 1. Determine the values of a and b.