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Y \ i y o f b $ f a x # \ a i g h $ !< 9 8 ;1) = f(g 2) Proof For all g 1, g 2 2G, we have g 1 kerf = g 2 kerf ()g 1 2 g 1 2kerf ()f(g 1 2 g 1) = e L ()f(g 2) 1f(g 1) = e L ()f(g 1) = f(g 2) The lemma tells us there is a bijective correspondence between the factor group G kerf and the image Imf In the next theorem, we put this to use to help us determine what can possibly be a

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F(x)d (a,b) 7!(a,bd) shift up by d f(x)d (a,b) 7!(a,bd) shift down by d cf(x) (a,b) 7!(a,cb) stretch vertically by c 1 cf(x) (a,b) 7!(a, 1 cb) shrink vertically by 1 c f(x) (a,b) 7!(a,b) flip over the xaxis Examples • The graph of f(x)=x2 is a graph that we know how to draw It's drawn on page 5946 Bijections and Inverse Functions A function f A → B is bijective (or f is a bijection) if each b ∈ B has exactly one preimage Since "at least one'' "at most one'' = "exactly one'', f is a bijection if and only if it is both an injection and a surjection A bijection is also called a onetoone correspondenceFind (f g)(x) for f and g below f(x) = 3x 4 (6) g(x) = x2 1 x (7) When composing functions we always read from right to left So, rst, we will plug x into g (which is already done) and then g into f What this means, is that wherever we see an x in f we will plug in g That is, g acts as our new variable and we have f(g(x)) 1

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Gf(x) = g(f(x)) This is sometimes called a function of a function An alternative notation for gf is g f Exercises 1 Work out gf(x) for the following pairs of functions (a) f(x) = 3x, g(x) = 2x2 − 5, (b) f(x) = e4x, g(x) = √ x, (c) f(x) = sinx, g(x) = 1/x wwwmathcentreacuk 2 c mathcentre 0922 3 Continuous Functions If c ∈ A is an accumulation point of A, then continuity of f at c is equivalent to the condition that lim x!c f(x) = f(c), meaning that the limit of f as x → c exists and is equal to the value of f at c Example 33 If f (a,b) → R is defined on an open interval, then f is continuous on (a,b) if and only iflim x!c f(x) = f(c) for every a < c < b(!(!(!(!(!(!(!(C r a t e r s e o f t h e M o o n W i l d e r n s s I d a h o a N a t i o n a l L a b

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9 1 / ;Fog or F composite of g (x) means plugging g (x) into f (x) An online gof fog calculator to find the (fog) (x) and (gof) (x) for the given functions In this online fog x and gof x calculator enter the f (x) and g (x) and submit to know the fog gof function Fog and Gof are the function composites or the composite functions f o g means Fcomposeg of x written as (f o g) (x) or f (g (x)), and G o f means Gcompose of g written as (g o f) (x) or g (f (x= 4 > 8 ?

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FOL Semantics (6) Consider a world with objects A, B, and C We'll look at a logical languge with constant symbols X, Y, and Z, function symbols f and g, and predicate symbols p, q, and rIf f(x,y) is convex in (x,y) and C is a convex set, then g(x) = inf y∈C f(x,y) is convex examples • f(x,y) = xTAx2xTBy yTCy with A B BT C 0, C ≻ 0 minimizing over y gives g(x) = infy f(x,y) = xT(A−BC−1BT)x g is convex, hence Schur complement A−BC−1BT 0 • distance to a set dist(x,S) = infy∈S kx−yk is convex if S is convex

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