This document provides a tutorial on calculus concepts including: - Identifying functions from graphs - Determining if graphs represent functions - Calculating function values and composite functions - Finding inverse functions - Solving for unknown functions given other related functions The tutorial covers functions, inverse functions, composite functions, and applying calculus concepts like evaluating functions and finding unknowns.

1. PMB 3004: Calculus
Tutorial 1
1. Which of the following illustrates a function?
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   
  
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  
  
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(a) (c)
   
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  
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2. State whether the following graphs represent functions or not. Give reasons for
your answers.
y
y
0 x
0 x
y
y
0 x
0 x

2. y
0 x
3. If f : x  2 x  3 , what is the image of 6 ? Find the values of f (6) , f (1) , f (0) ,
5
f (12) .
4. If f ( x)  2 x  3 and g ( x)  x 2  5 , find the following:
(a) gf (2) ; (b) fg (3) ; (c) fg (a  1) ; (d) gf ( x  1) .
5. If f : x  2 x , g : x  x  1 and h : x  x , build up formulae for the composite
2
functions fgh , hfg , gfh and hgf .
6. Find the inverse functions for each of the following:
(a) f : x  x  4 ; (b) g : x  3x  1 ; (c) h : t  12 - 7
4 t
7. Given that f : x  x , find the value(s) of x for which f 1 ( x)  f ( x) .
2x  3
8. Find f if g and gf are given as follows:
(a) g(x) = 2/x , x ≠ 0 and gf(x) = 2/(x-1), x ≠ 1
(b) g(x) = x-2 and gf(x) = 3x-7
9. Find g if f and gf are given as follows:
(a) f(x) = 4x , x ≠ 0 and gf(x) = 4x-3
(b) f(x) = 2x-1 and gf(x) = 6x-3