NJC 2017 H2 Maths Exponential Logarithm and Modulus Functions and their Graphs Notes

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 1 of 9 National Junior College 2016 – 2017 H2 Mathematics Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Key Questions to Answer:  What is an exponential function?  What is the range of values for which an exponential function is well-defined  What laws do exponential functions follow?  What does the graph of an exponential function look like?  What are the key characteristics of the graph of an exponential function?  What is a logarithmic function?  What is the range of values for which a logarithmic function is well-defined?  What laws do logarithmic functions follow?  What does the graph of a logarithmic function look like?  What are the key characteristics of the graph of a logarithmic function?  What is the modulus function?  When do we use the modulus function?  How do I manipulate the modulus function?  How can I draw a graph of a modulus function? §1 Exponential Functions Definition 1.0.1 (Exponential Function) A function , 0 1,xy a a a  , is known as an exponential function . It is a function used to model a relationship for which a constant increase in the independent variable (here denoted by x) gives the same proportional change in the dependent variable (here denoted by y). The most common exponential function is exy . UNDERSTAND What is meant by ‘increasing exponentially’?

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 2 of 9 1.1 Laws of Indices If , , , , 0, 0a b m n a b   (i) mn a a = mna  (ii) mna a = mna  (iii)   nma  mna (iv)   mmma b ab (v) m m a b  m a b   (vi) 1 0n n a , aa  (vii) 0 10a , a (viii) 1/n naa  (ix)   m mn /n aa mna (x) , where 1,0,1xn a x na a    WONDER Why is that for (x), 1,0,1a ? Exercise 1 Without the use of a calculator, simplify the following expressions: (a) 11 369 9 (b) 12 24 2 n n n   (c) 1 0.528 2 (d) 3312 6 (a) 11 1 36 299 9 3   (b) (c) 1 3 1 0.5 22 2 28 2 42 2     (d)

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 3 of 9 1.2 Graph of the Exponential Function Key features of the graph of the exponential function: Axial Intercepts  0,1 Asymptotes 0y Exercise 2 Identify the key features of the graphs of the following exponential functions and sketch them. (Hint: use your graphic calculator) (a) 2, 1xya a  (b) 23 xy (c) e1xy   (d) 21e1xy   (a) (b) (c) (d) WONDER How should the graph of xya look like if 01 a ? y x y x 2 1 y x 1 Asymptote at 0y ,1xy a a

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 4 of 9 §2 Logarithmic Functions Definition 2.0.1 (Logarithmic Function) A function of the form logay x , where a , 0a , 1a and 0x . loga x is read as ‘the logarithm of x to base a’ or more simply, ‘log, base a, x’. Of special importance will be those with base e, i.e. those of the form e llog nxy x . WONDER What are some practical uses for logarithmic functions? 2.1 Laws of Logarithms If , , , ,a b c x y  + and r , (i) log log loga a axy x y (ii) log log loga a a x xyy   (iii) log logr aax r x (iv) loglog log c a c bb a (v) 1log 1a a aa  (vi) 0log 1 0 1a a  Take note that (i) is not equivalent to saying log ( ) log loga a ax y x y   , In fact, log ( ) log loga a ax y x y   .

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 5 of 9 Exercise 3 Simplify and express the each of the following as a single logarithm. (a) 5 3lo2 glog 2 log 4x x x (b)       22lg 2 lg 1 lg 32x x x x   (c) 3 2lg5 (d) 324l3log ogaa a (a) 25 252l 5 3log 2 log 4 log log 82og 4x x x x x               (b) (c) 310 10003 2lg5 3lg10 lg 25 lg lg lg 4025 25      (d)

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 6 of 9 2.2 Graph of the Logarithmic Function Key features of the graph of the logarithmic function: Axial Intercepts  1,0 Asymptotes 0x For  logay bx c , graph does not exist for 0bx c  Exercise 4 Identify the key features of the graphs of the following logarithmic functions and sketch them: (a)  ln 2 1yx (b)  2ln 1yx   (c)  2log 1yx (d)  ln 2 3 1yx   (a) (b) (c) (d) WONDER What do you observe about the graphs in Exercise 4 as compared to that in the graph at the top of the page? y x 0.5x 0 y x 1x 0 y x 0 Asymptote at 0x  2logyx

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 7 of 9 §3 Modulus Functions Definition 3.0.1 (Modulus Function) The absolute value or modulus of a real number x is denoted by x . Formally, ,0 ,0 xxx xx  UNDERSTAND What are some alternative interpretations of the modulus function? WONDER The modulus function is an example of a piece -wise function. Can you think of any other piece-wise functions? 3.1 Properties of the Modulus Function For all ,, xy (i) 0,x (ii) xy x y . Hence, (a) 1   x x x (b) times times      nn nn x x x x x x x x for any positive integer n, (iii) or x y x y x y    (iv) For a general function f(x), f( ) if f( ) 0f( ) f( ) if f( ) 0  xxx xx CHECK Is the following correct? f( ) if 0f( ) f( ) if 0  xxx xx In general, , or , where 0x k x k x k k     , and or a b a b a b    . EXPLORE How can we apply the definition of the modulus function to solve inequalities involving the modulus function?

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 8 of 9 Exercise 5 Solve the following equations. (a) |x – 6| = 7 (b) |x2 – 5x – 1| = 5 (c) |x2 – 5x + 1| = – 5 Solution: (a) |x – 6| = 7  x – 6 = 7 or x – 6 = – 7  x = 13 or x = – 1 (b) (c)

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 9 of 9 3.2 Graph of the Modulus Function (y = |f(x)|) Notice that when 0x , the graph of y x is the same as that of yx , and when 0x , the graph of y x is the same as that of yx , which agrees with the definition of x . In general, for any curve y = f(x), (i) the curve y = – f(x) is a reflection of y = f(x) about the x-axis. (ii) the curve y = |f(x)| is obtained by keeping the part of the graph of y = f(x) that is above the x-axis, and reflecting the part of the graph of y = f(x) below the x–axis about the x–axis. Exercise 6 Sketch the graph of y = |f(x)| for the following graph of y = f(x).