No They have no special significance All of these are simply used by convention to refer to a function For example, f(x)=e^x, a(x)=e^x, and z(x)=e^x all mean the same thing However, some notations mean only one thing For example, zeta(s) refers to the Riemann zeta function sum_(n=1)^oo1/n^s (See this link for more info Riemann zeta function) In general,Thus, f first translates x into I, if it is outside I, and otherwise, untranslates and computes g, if it is in I It follows that f (f (x)) = g (x) for all x outside I There are 2 R many such h's, and hence also this many f'sQED If g is continuous, then this f can be chosen also to be continuousF(x) = x 3 g(x) = x 5 defined for all real numbers (Note composite functions may have a different notation such as (f g)(x)) Q1) Solve the equation fg(x) = 27 Firstly, we must find the composite function fg(x) in terms of x before we solve it In order to do this, we can break down the function in the following way fg(x) = fg(x) = f(x5)
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What does (f•g)(x) mean-(a) For any constant k and any number c, lim x→c k = k (b) For any number c, lim x→c x = c THEOREM 1 Let f D → R and let c be an accumulation point of D Then lim x→c f(x)=L if and only if for every sequence {sn} in D such that sn → c, sn 6=c for all n, f(sn) → L Proof Suppose that lim x→c f(x)=LLet {sn} be a sequence in D which converges toc, sn 6=c for all nLet >0When you find (f o g)(x), there are two things that must be satisfied x must be in the domain of g, which means x is a real number (pretty easy to do) g(x) must be in the domain of f, which means that 1x 2 ^2 ≥ 4 (when you try to solve this, you get the empty set);
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G (x) =2x or h (x) =2x,,,,,mean the same thing,,,except the axis is now g (x),,or h (x) The advantage of using functional notation is that different items can be differentiated, and still shown to be a function of x If we had a cost, we might use c (x) If we just used "c", it might not be clear that it is a function of xVertical Translation For the base function f (x) and a constant k, the function given by g (x) = f (x) k, can be sketched by shifting f (x) k units vertically Horizontal Translation For the base function f (x) and a constant k, the function given by g(x) = f (x k), can be sketched by shifting f (x) k units horizontally Vertical Stretches and ShrinksThe following rules apply to any functions f(x) and g(x) and also apply to left and right sided limits Suppose that cis a constant and the limits lim x!a f(x) and lim x!a g(x) exist (meaning they are nite numbers) Then 1lim x!af(x) g(x) = lim x!af(x) lim x!ag(x) ;
Therefore, we can conclude that A is open, meaning that its complement, {xf(x) ≤ g(x)} is closed (b) Let h X → Y be the function h(x)min{f(x),g(x)} Show that h is continuous Proof By a similar argument to that made in (a) above, we can show that B = {xg(x) ≤ f(x)} is closed Also, since f and g are continuous on X, it is true that In order to prove f(x) = O(g(x)), we need to find two positive constants, c and x 1, such that 0 ≤ f(x) ≤ cg(x) for all x ≥ x 1 We need to find values for c and x 1 such that the inequality holds What it means, is that past a certain point, a scaled version of g(x) will always be bigger than f(x) 6 Example 2*f(x) means two multiplied by the function f f(2x) means the function at 2x;
H(x) = f (x)g (x) h ( x) = f ( x) g ( x) Since f (x)gx f ( x) g x is constant with respect to f f, the derivative of f (x)gx f ( x) g x with respect to f f is 0 0 0 0 So what does this mean (f g)(x), the composition of the function f with g is defined as follows (f g)(x) = f(g(x)), notice that in the case the function g is inside of the function f Whereas in the composite(g f)(x), g(x) is the outside function and f(x) is the inside functionFor example, the function g (x) = f^ (1/2) (x) would be a function that satisfies g^2 (x) = f (x) Also, as a side note, the neutral function is more commonly called the identity function (and the neutral element 1 is called the identity element) (The trigonometric functions break this convention sin^2 (x) is taken to mean sin (x)*sin (x)
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1 Introduction The composition of two functions g and f is the new function we get by performing f first, and then performing g For example, if we let f be the function given by f(x) = x2 and let g be the function given by g(x) = x3, then the composition of g with f is called gf and is worked out(the limit of a sum is the sum of the limits) 2lim x!af(x) g(x$\begingroup$ Does "by definition" mean by the limit definition?
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The output f (x) is sometimes given an additional name y by y = f (x) The example that comes to mind is the square root function on your calculator The name of the function is \sqrt {\;\;} and we usually write the function as f (x) = \sqrt {x} On my calculator I input x for example by pressing 2 then 5 Then I invoke the function by pressingIf f(x) and g(x) are differentiable functions, then the derivative of the composition of g with f is where the notation g'(f(x)) means the function g'(x) evaluated at f(x) Once again, this result can be established from the definitionOr the value of the function evaluated at 2x Giving a name f to a function for the function using independant variable x will be named as f(x), to be read, "the function f of x" Shown alone, f and x are not factors, but are a complete name
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For f(x) into the formula for g(x) g(f(x)) = g p 4 x2 = 1 (p 4 x2)2 4 = 1 (4 x2) 4 = 1 2x Now, to nd the domain of (g f)(x), we consider both the domain of 1 2x and the domain of f(x) The domain of 1 2x includes every value except x= 0, and written in interval notation is (1 ;0)(0;1) The domain of f(x) = p 4 x2 is determined by nding where A more common notation is f = Θ (g (x)) (see wikipedia), but as the latter is a set of functions, a more settheoretical notation is to write f ∈ Θ (g (x)) instead It says that f belongs to a certain set of functions visàvis gThe Domain of g (x) = x2 is all the Real Numbers The composed function is (g º f) (x) = g (f (x)) = (√x)2 = x Now, "x" normally has the Domain of all Real Numbers but because it is a composed function we must also consider f (x), So the Domain is all nonnegative Real Numbers
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$\endgroup$ – user Nov 21 '16 at 1656 $\begingroup$ Ahh, my bad!• Constant Multiple Rule g(x)=c·f(x)theng0(x)=c·f0(x) • Power Rule f(x)=x n thenf 0 (x)=nx n−1 • Sum and Difference Rule h(x)=f(x)±g(x)thenh 0 (x)=f 0 (x)±g 0 (x) Problem Assume that f has a derivative everywhere Set g(x)=xf(x) Using the definition of the derivative, show that g has a derivative and that g'(x)=f(x)xf'(x) What I know I know the definition of the derivative is f(xh)f(x)/h I don't know how to plug it
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