1. f: Z-Z decide whether has a right inverse or a left and exhibit these inverses if f has. (1) f(x)=x-1 (2)(x)=1+x (1+x If xin even if x is odd 2
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Nn.100.
Subject :- Advance Mathematic
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- Suppose f is an invertible function and f^-1 is its inverse. Explain how we can use the chain rule to differentiate f(f^−1(x)). Note: f^-1(x) is not 1/f(x)we have f: X→Y is a 1-1 function and Y is countable. Since f is 1-1, is it correct that this implies X ~ f(X) [which is f:X →f(X)] which further implies bijecton? Also, does this imply f(X) ⊆ Y?Let f, g: R→R be defined byf (x) = {4x + 1 , if x ≥0, x , if x < 0} g(x) = {3x , if x ≥0 , x + 3 , if x < 0.} Find the inverse of f ◦g.
- Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval. f(x) = ∣x + 2∣, [−2, ∞)The function f : R → R defined by the formula f(x) = 3x − 5 for all x ∈ R was shown to be one-to-one in Example 2 and onto in Example 4. Find its inversefunction.Show also that the function ln(z) is unique up to addition of a constant 2πik, k ∈ Z.
- Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval. f(x) = (x − 4)2, [4, ∞)If f is a function of only x, g is a function of only y, and f and g satisfyln(f(x)) + ln(g(y)) = ln(1 + x + y + xy),find f(x) and g(y).Can we explicity "prove that f(x)=x^{1/k} can be defined on [0,\infty) by the requirement that it be the inverse function of g(x)=x^k on [0,\infty), where k is any positive integer"?
- 1. If f (x) = |x − 1| + |x + 3|, then discuss the continuity and differentiability of the function at x = −3 and x = 1.The maximum value of F in F(x) = - 4 cos x is 2 4 1b. Consider the compound function: (x - 2)3 for x less than or equal to 2h(x) = |x| for 2 < x < 5 x2 – 10 for x greater than or equal to 5 Is the function h(x) continuous everywhere over the interval (2 , 5) and is the function h(x) continuous everywhere on the Real Line?