Functions and their Combinations Machine f f(x) = x2 x 1 2 4 3 9 -2 4 -1 1 -3 9 f(x) A Basic Simple Function

Sum of two functions: (f+g)(x) Machine f f(x) = x2 x (f+g)(x) + Machine g g(x) = 2x f(x)+g(x) x f(x) g(x) f(x)+g(x) 1 1 2 3 2 4 4 8 What is the domain and range of this combination function? We will discuss that soon. 4 16 8 24 10 100 20 120

Example of Sum (or Difference) of Functions Let f(x) = 2x – 1; let g(x) = 4x +12 Then (f + g)(x) = f(x) + g(x) = 2x –1 + 4x + 12 = 6x + 11 Also (f – g)(x) = f(x) – g(x) = 2x – 1 – (4x + 12) = -2x – 13 Evaluate (f + g)(4) = f(4) + g(4) = 2x –1 + 4x + 12 = 6x + 11 = 6(4) + 11 = 35

Product of two functions:f(x)*g(x) Machine f f(x) = x2 x (f*g)(x) * Machine g g(x) = 2x f(x)*g(x) x f(x) g(x) f(x)*g(x) 1 1 2 2 2 4 4 16 Do you think the range of this combination function is limited? We will see soon. 4 16 8 128 10 100 20 2000

Example of Product of Functions Let f(x) = 2x – 1; let g(x) = 4x +12 Then (f * g)(x) = f(x) * g(x) = (2x – 1) * (4x + 12) = 8x2 + 20x – 12 Note also: (g * f)(x) = g(x) * f(x) = (4x + 12)*(2x – 1) = 8x2 + 20x – 12 Evaluate (f * g)(1) = f(1) * g(1) = (2x – 1) * (4x + 12) = 8x2 + 20x – 12 = 8(1)2 + 20(1) – 12 = 16

Quotient of two functions: (f/g)(x) *Provided g(x) 0 Machine f f(x) = x+1 x (f/g)(x)  Machine g g(x) = 2x f(x)/g(x) x f(x) g(x) f(x)/g(x) What does undefined mean? It means there is no answer 1 2 2 1 2 3 4 3/4 6 7 12 7/12 1 Undefined

Example of Quotient of Functions Let f(x) = x2 – 1; let g(x) = x – 1 Then f(x) / g(x) = (x2 – 1) / (x – 1) = [(x + 1)(x – 1)] / (x – 1) = x + 1 except when x = 1 because then g(x) is zero So the domain of (x2 – 1) / (x – 1) is all Real numbers except for exactly 1. More precisely -  < x < 1 OR 1 < x < . But x cannot be exactly one. More on this soon.

What does ‘undefined’ mean? Do you believe that 2*3 = 6? That also means that 6/3 =2! So what does 6/0 mean?? 6/0 = ???? 6/0 means there must be some number ‘????’ such that: 6 = ??? * 0 ! So what number times 0 equals 6??? There is no answer. Dividing by zero is ‘undefined’! Try dividing by zero on your calculator; it gives an error So the quotient of two functions is only possible when the divisor is not equal to zero

Undefined is infinity? Another way to think about dividing by zero is that the answer is infinity. But infinity is not an actual number. If you think about zero as pretty close to a really tiny number like 0.000000000001 that is: 1/1000000000000 You might also recall that dividing by a fraction is the same as ‘multiplying by its reciprocal’. So a number divided by a very very tiny number is huge So dividing by zero can, in some contexts be considered to be infinity! But be aware, infinity is not a number!

‘Composition’ of functions The composition of a function is denoted as: It means ‘f(g(x))’, which means the value of g(x) machine is fed into the f(x) machine! Machine g g(x) = x2 x g(x) Machine f f(x) = g(x) + 1 f(g(x)) The output of this machine is really x2 + 1; because the ‘f’ stage treats the x2 output of the ‘g’ stage as an input x to the f machine. (fg) = f(g(x)) = f(x2) = x2 + 1

Composition of a function Machine g g(x) = x2 x g(x) Machine f f(x) = g(x) + 1 f(g(x)) x g(x) f(g(x)) 1 1 2 Do you think the range or domain of this may have some limits? 2 4 5 6 36 37 8 64 65 Do you think f(g(x)) = g(f(x)) ?

Composition of Functions Examples f(g(x)) a) Let f(x) = x + 3; let g(x) = x Then f(g(x)) = x+3 but x must be > 0 limits the domain since you can’t take the  of a negative number. So the range will be f(g(x))  3. b) Let f(x) = 1 / x; let g(x) = x2 + 4 Then f(g(x)) = 1 / (x2 + 4) but x may not equal 2 or –2 so the domain is limited. Further the range of f(g(x)) will always be > 1 / 4

f(g(x)) is not necessarily equal to g(f(x)) Machine g g(x) = x2 x g(x) Machine f f(x) = g(x) + 1 f(g(x)) = x2+1 Machine f f(x) = x + 1 x f(x) Machine g g(x) =(f(x))2 g(f(x)) = (x+1)2 =x2+2x+1 Can you think of a case where f(g(x)) = g(f(x))?

Many complex functions can be thought of as compositions