1.7 Combination of Functions Add, Subtract, Multiply and Divide functions Understanding combination functions

Sum and Difference Let f(x) = 2x + 5 and g(x) = x2-3 Sum f(x) + g(x) = 2x + 5 + x2-3 thus (f + g)(x) = x2 + 2x + 2 Difference (f - g)(x) = 2x + 5 – (x2 – 3) = -x 2 + 2x + 8

Product and Quotient Let f(x) = 2x + 5 and g(x) = x2-3 Product f(x)g(x) so (fg)(x) = (2x + 5)(x2 – 3) = 2x3 + 5x2 – 6x – 15 Quotient g(x)≠0

So what is the domain of the Quotient function?

So what is the domain of the Quotient function? All reals that does not make the denominator zero. What about a square root?

Let and The Domain of is [0, ∞) The Domain of [ - 5, 5 ] Domain is [0, 5) ; Why?

The zero comes from can not have negative numbers, but must have numbers less then 5. Since it is in the denominator g(x) can not be zero. How would the Domain change if The Domain would be (0, 5] What cause the difference?

Composition of Function Composition of function is where the range of one function become the domain of the other function. Let f(x) = x3 + 1 and g(x)= x + 4 Old way of written a composition was f(g(x)) New way (f∘g)(x) = (x+4)3 + 1 (g∘f)(x) = (x3+1)+4

(f∘g)(x) = (x+4)3 + 1 (x + 4)3 = (x + 4)(x + 4)(x + 4) =(x2 + 8x + 16)(x + 4) =(x3 + 8x2 + 16x)+(4x2 + 32x + 64) = x3 + 12x2 + 48x + 64 (f∘g)(x) = x3 + 12x2 + 48x + 64 + 1 = x3 + 12x2 + 48x + 65

Let f(x)= x2 + 8 and Domain of f(x) is ( - ∞,∞) Domain of g(x) is [- 4, 4] So the Domain would domain in which both equations work. [ -4, 4]

Why use Composition functions? To break functions into smaller easier to handle parts. h(x)= (f∘g)(x) Into 2 equations f(x) = 1/x and g(x) =(x+ 3)2 or f(x) = (x + 3)-2 and g(x) = (x + 3)

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