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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