Math 1A: introduction to functions and calculus Oliver Knill, 2011Lecture 2: FunctionsA function is a rule which assigns to a real number a new real number. An exampleis f(x) = x2−x. For example, it assigns to the number x = 3 the value 32−3 = 6. Afunction is given with a domain A, the points where f is defin e d and a codomainB a set of numbers in which f takes values.Typically, t h e codomain agrees with the set of real numbers and the d om ai n to be all the numbers,where the function is defined. The function f( x) = 1/x for example is not defined at x = 0 so thatwe chose the domain A = R \{0}, all numbers except 0. The function f(x) = 1/x takes values inthe codomain R. If we choose A = B, then f(x) = 1/x reaches every point in B and is invertible.It is its own inverse. Here are a few examples of functions. We will look at them in more detailduring the lecture, especially the polynomials, trigonometric functions and exponential funct i on .identity f(x) = xconstant f(x) = 1linear f(x) = 3x + 1quadratic f(x) = x2cosine f(x) = cos(x)sine f(x) = sin(x)exponentials f(x) = exph(x) = (1 + h)x/hlogarithms f(x) = logh(x) = exp−1hpower f(x) = 2xexponential f(x) = ex= exp(x)logarithm f(x) = log ( x) = exp−1(x)absolute value f(x) = |x|devil comb f(x) = sin(1/x)bell function f(x) = e−x2witch of Agnesi f(x) =11+x2sinc sin(x)/xWe can build new functions by:add functions f(x) + g(x)scale functions 2f(x)translate f(x + 1)compose f(g(x))invert f−1(x)difference f(x + 1) − f(x)sum up f(x) + f(x + 1) + . . .Here are important functions:polynomials x2+ 3x + 5rational functions (x + 1)/(x4+ 1)exponential exlogarithm log(x)trig functions sin(x), tan(x)inverse trig functions arcsin−1(x), arctan(x).roots√x, x1/3We will look at these functions a lot during t h i s course. The logarithm, exponential and trigono-metric functions are especially important.For some functions, we need to restrict the domain, where the function is d efi ned. For the squareroot fun ct i on√x or the logarithm log(x) for example, we have to assume t h a t the number ispositive. We write that the domain is (0, ∞) = R+. For the function f (x) = 1/x, we have toassume that x is different from zero. Keep these th r ee examples in mind.The graph of a function is the set of points {(x, y) = (x, f(x)) } in the plane, wherex runs over the domain A of f. Graphs allow u s to visualize fun ct i ons. We can”see them”, when we draw the g r ap h.expHxLxlogHxLxe- x2xx sinH1xLxxxx3- 3 xxHomework1 Draw the function f(x) = x + sin(x). Its graph goes through the origin (0, 0).a) A function is called odd if f(−x) = −f(x). Is f odd?b) A function is called even if f(x) = f(−x). Is f even?c) A function is called monotone increasing if f(y) > f (x) if y > x. Is f monotoneincreasing? You do not have to decide this yet analy t i cal l y. Just draw(∗)the function andmake up your mind.2 A function f : A → B is called invertible or one to one if there is an other functiong such that g(f(x)) = x for all x in A and f(g(y)) = y for all y ∈ B. For example, thefunction g(x) =√x is the inverse of f( x) = x2as a function from A = [ 0, ∞) to B = [0, ∞).Determine from the fol l owing functions whether they are invertible. If they are invertible,find the inverse.a) f(x) = sin(x) from A = [0, π/2] to B = [0, 1]b) f(x) = x3from A = R to B = Rc) f(x) = x6from A = R to B = Rd) f(x) = exp(5x) from A = R to B = R+= (0, ∞).e) f(x) = 1/(1 + x2) from A = [0, ∞) to B = [0, ∞).3 Look at the fu nction f1(x) = sin(x), f2(x) = sin(sin(x)), f3(x) = sin(sin(sin(x))).a) Draw the graphs of the funct i ons f1, f2, f3on the interval [0, 4π].b) Can you imagine what f100000(x) looks like? You might want to make more experimentshere to see the answer. Of course you are allowed to plot the functions with a calculator orwith an online grapher like Wolfram alpha. (The weblink can be found below) .4 Let s call a function f (x) a composition square root of a function g if f(f(x)) = g( x ) . Forexample, the funct io n f(x) = x2+ 1 is the composition square root of g(x) = x4+ 2x2+ 2because f (f(x)) = (x2+ 1)2+ 1 = g(x). Find the composition square roots of the followingfunctions:a) f(x) = sin(sin(x)).b) f(x) = x4c) f(x) = xd) f(x) = x4+ 2x2+ 2e) f(x) = eex.Note that it can be difficult in gene ra l to fi nd the squar e root function in general. Alreadyfor basic functions like exp(x ) or sin(x), we are speechless.5 A function f (x) has a root at x = a if f(a) = 0. Roots are places, wher e the functi on iszero. Find one root for each of the following function s or state that there is none.a) f(x) = sin(x)b) f(x) = exp(x)c) f(x) = x3− xd) f(x) = sin(x)/x − 1e) f(x) = csc(x) = 1/ sin(x)(*) Here is how you can use t h e Web to plo t a function. The example given is sin(x).✞http : / /www. wolframalpha . com/ in put /? i=Plot+s i n ( x )✝
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