By Hans Sagan

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**Extra resources for Advanced calculus: Of real-valued functions of a real variable and vector-valued functions of a vector variable**

**Example text**

S,. (2) Let (xl, yl) be a point on a line with slope m. Then if ( x , y ) is any other point on the line, its coordinates must satisfy, from ( 2 ) , y - y1 = m ( x - (3) XI). Equation (3) is called a point-slope equation of the line. EXAMPLE 1 Find a point-slope equation of the line passing through the points ( - 1, - 2 ) and (2,5). Solution. We first compute m = 5 - ( - 2 ) -- _7 2 - (-1) 3' Thus if we choose ( x l , yl) = (2, 5 ) , a point-slope equation of the line is y - 5 = ;(x - 2). Choosing (x,, yl) y - y +2 (-2) = = ( - 1, - 2 ) , we obtain another point-slope equation of the line: ; [ x - (-1)]/ or = <(x + 1).

Note that R ° R = } = SoS. (a) Let Γ = R ° S. Show that T ° Γ = S ° R and T ° T ° T = /. (b) Let tf = R o S o β. Show that ii = S ° R ° S and li ° Ü = /. R°r°r = r ° r ° s = s ° r . (d) Show that the set of six functions {/, R, S, T, T ° Γ, U} is closed with respect to the operation of composition; that is, if F and G belong to the set, then so does F ° G. (e) Show that for each function F in the set {/, R, S, T, T ° T, U} there is a unique function G in the set such that F°G = J=G°F. REMARK. This set of functions forms what is called a group.

B) g ( / ( 0 ) = * forali* E [0, «>). 32. Let / and g be the following straight-line functions: fix) = ax + b, gix) = ex + d. Find conditions on a and fr in order that / ° g = g ° / . *33. For any two subsets A and B of dorn / , show the following: (a) fiA U B ) = /(A) U /(B) (b) /(Λ f i ß ) C /(Λ) Π /(B) 34. Each of the following functions satisfies an equation of the form if ° /)(x) = x or ig ° g ° g)(x) = x or (/z ° /z ° h ° /i)(x) = x, and so on. For each function, discover what type of equation is appropriate.