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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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 ) = x 2 + 2x - 1 find ( f + g ) ( x ) and ( f + g ) ( 2 )Solution
Step 1. Find ( f + g ) ( x )
Since ( f + g ) ( x ) = f ( x ) + g ( x ) then;
( f + g ) ( x ) = ( 2x + 1 ) + (x 2 + 2x - 1 ) = 2x + 1 + x 2 + 2x - 1 = x 2 + 4xStep 2. Find ( f + g ) ( 2 )
To find the solution for ( f + g ) ( 2 ), evaluate the solution above for 2.Since ( f + g ) ( x ) = x
2 + 4x then; ( f + g ) ( 2 ) = 2 2 + 4(2) = 4 + 8 = 12 Example 2. Given f ( x ) = 2x - 5 and g ( x ) = 1 - x find ( f ) ( x ) and ( f ) ( 2 ).Solution
Step 1. Find ( f - g ) ( x ).
( f ) ( x ) = f ( x ) ( x ) = ( 2x - 5 ) - ( 1 - x ) = 2x - 5 - 1 + x = 3x - 6Step 2. Find ( f ) ( 2 ).
( f ) ( x ) = 3x - 6 ( f ) ( 2 ) = 3 (2) - 6 = 6 - 6 = 0Example 3. Given f ( x ) = x
2 + 1 and g ( x ) = x - 4 , find ( f g ) ( x ) and ( f g ) ( 3 ).Solution
Step 1. Solve for
( f g ) ( x ).Since ( f g ) ( x ) = f ( x ) * g ( x ) , then
= (x 2 + 1 ) ( x - 4 ) = x 3 - 4 x 2 + x - 4 .Step 2. Find ( f g ) ( 3 ).
Since ( f g ) ( x ) = x
3 - 4 x 2 + x - 4, then ( f g ) ( 3 ) = (3) 3 - 4 (3) 2 + (3) - 4 = 27 - 36 + 3 - 4 = -10 Example 4. Given f ( x ) = x + 1 and g ( x ) = x - 1 , find ( x ) and ( 3 ). f g f gSolution
Step 1. Solve for ( x ).
f gSince ( x ) = , then
fx gx f g = ; x 1 1 1x xStep 2 Find .
3f gSince = , then
1 1x x f x g 3131
3f g 4 2 = 2 NOTE: Any restrictions on the domains of f or g must be taken into account when performing these operations. Composition is another operation that may be performed among functions. Simply stated, it is evaluating one function in terms of another. The format for composition is: (f g)(x) = f(g(x)).