Graph Transformations - University of Utah
Notice that all of the “new functions” in the chart di↵er from f(x)bysome algebraic manipulation that happens after f plays its part as a function For example, first you put x into the function, then f(x) is what comes out The function has done its job Only after f has done its job do you add d to get the new function f(x)+d 67
The Algebra of Functions - Alamo Colleges District
The Algebra of Functions Like terms, functions may be combined by addition, subtraction, multiplication or division Example 1 Given f ( x ) = 2x + 1 and g ( x ) = x2 + 2x – 1 find ( f + g ) ( x ) and
WORKSHEET &# 8 IRREDUCIBLE POLYNOMIALS
Then f is irreducible if and only if f(a) 6= 0 for all a2k Proposition 0 4 Suppose that a;b2kwith a6= 0 Then f(x) 2k[x] is irreducible if and only if f(ax+b) 2k[x] is irreducible Theorem 0 5 (Reduction mod p) Suppose that f2Z[x] is a monic1 polynomial of degree >0 Set f p 2Z modp[x] to be the reduction mod pof f (ie, take the coe cients
Composition Functions
Find (f g)(x) for f and g below f(x) = 3x+ 4 (6) g(x) = x2 + 1 x (7) When composing functions we always read from right to left So, rst, we will plug x into g (which is already done) and then g into f What this means, is that wherever we see an x in f we will plug in g That is, g acts as our new variable and we have f(g(x)) 1
AP 2006 Calculus AB Form B scoring guidelines
first derivative of f, given by f ′()xe x= ()−x 4 sin ()2 The graph of yfx= ′() is shown above (a) Use the graph of f ′ to determine whether the graph of f is concave up, concave down, or neither on the interval 1 7 1 9
AP CALCULUS AB 2014 SCORING GUIDELINES
f x dx Thus, if the vertical line x k = divides R into two regions with equal areas, then ( ( )) 2 3 0 4 4 k k
CalculusReview [328 marks]
Part of the graph of f is shown in the following diagram The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 Find the volume of the solid formed when R is revolved 360° about the x-axis
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
1: F(-∞)= 0 and F(∞)=1; 2: If a < b, then F(a) ≤ F(b) for any real numbers a and b 1 6 3 First example of a cumulative distribution function Consider tossing a coin four times The possible outcomes are contained in table 1 and the values of p(·) in equation 2 From this we can determine the cumulative distribution function asfollows
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WORKSHEET # 8
IRREDUCIBLE POLYNOMIALS
We recall several dierent ways we have to prove that a given polynomial is irreducible. As always,kis a eld.
Theorem 0.1(Gauss' Lemma).Suppose thatf2Z[x]is monic of degree>0. Thenfis irreducible inZ[x]if and only if it is irreducible when viewed as an element ofQ[x]. Lemma 0.2.A degree one polynomialf2k[x]is always irreducible.Proposition 0.3.Suppose thatf2k[x]has degree 2 or 3. Thenfis irreducible if and only iff(a)6= 0for all
a2k. Proposition 0.4.Suppose thata;b2kwitha6= 0. Thenf(x)2k[x]is irreducible if and only iff(ax+b)2k[x] is irreducible. Theorem 0.5(Reduction modp).Suppose thatf2Z[x]is a monic1polynomial of degree>0. Setfp2Zmodp[x]to be the reduction modpoff(ie, take the coecients modp). Iffp2Zmodp[x]is irreducible for some primep,
thenfis irreducible inZ[x].WARNING: The converse need not be true.
Theorem 0.6(Eisenstein's Criterion).Suppose thatf=xn+an1xn1++a1x1+a02Z[x]and also that there is a primepsuch thatpjaifor allibut thatp2doesNOTdividea0. Thenfis irreducible.1.Consider the polynomialf(x) =x3+x2+x+2. In which of the following rings of polynomials isfirreducible?
Justify your answer.
(a)R[x] (b)C[x] (c)Zmod2[x] (d)Zmod3[x] (e)Zmod5[x] (f)Q[x]Solution:
(a)It is reducible (= not irredu cible)b ecauseit is a cubic p olynomialand therefore has a ro ot. Thusfcan
be factored asf(x) = (x)g(x). (b) The ro otfrom (a) is also a complex n umber,and so fis reducible inC[x] as well. (c) Mo d2, f2=x3+x2+x, which has a root atx= 0 and so is reducible. (d) Mo d3, f3=x3+x2+x+2. 0 is not a root,f2(1) = 5 = 26= 0, and nallyf2(2) = 8+4+2+2 = 16 = 16= 0.