Chapter 1 Relations and Functions
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See all →- Let R → R be defined as f(x) = 10x + 7. Find the function g : R → R such that gof = fog = IR. Let f : X → Y, where X ⊆ R, Y ⊆ R. Consider an arbitrary element y of Y. By definition, y = 10x + 7 for some x ∈ X. ⇒ x = y-7 / 10 . Let us... 1
- Let f : W → W be defined as f(n) = n -1, if n is odd and f(n) = n + 1, if n is even. We are given, f : W → W, defined as \beginarrayll n-1, & n is odd \\ n+1, & n is even endarray. Let f(n1) =f(n2). If n1 is odd and n2 is eve... 1
- If f : R → R is defined by f(x) = x² – 3x + 2, find f[f(x)]. f(x) = x² – 3x + 2 f[f(x)] = [f(x)]² – 3[f(x)] + 2 = (x² – 3x + 2)² – 3(x² -3x + 2) + 2 = (x4 + 9x² + 4 – 6x³ – 12x + 4x²) – 3x² + 9x – 6 +... 1
- Show that the function f : R → { x ∈ R : – 1 – x < 1} defined by f(x) = \(\frac{x}{1+|x|}\) x ∈ R is one-one and onto. (i) Let x >0, f(x) = x / 1+x So, f(x1) = f(x2) ⇒ fracx₁1+x₁ = fracx₂1+x₂ or x1(1 + x2) = x2(1 + x1) or x1 + x1x2 = x2 + x1x2 ⇒ x1 = x2 When... 2
- Show that the function f : R → R given by f(x) = x³ is injective. f(x) = x³ ⇒ f(x1) = f(x2) ⇒ x³1 = x³1 ⇒ x1 = x2 ⇒ f is one-one, i.e., f is injective. 1
- Give examples of two functions f : N → Z and g : Z → Z such that gof is injective but g is not injective. Example: f(x) = x for f:mathbbN→mathbbZ , and g(x) = |x| for g:mathbbZ→mathbbZ . g is not injective: g(-1) = |-1| = 1 and g(1) = 1 , so two... 1