SAT II Math Level 2 Subject Test Notes: Functions

Functions Function Notation If  f and  f and g   g name name two functions, the following rules apply: 1. ( f +  f + g )( )( x  x)) = f  = f (( x)  x) + g  + g ( x)  x) 2. ( f   f  × g )( )( x  x)) = f  = f (( x)  x) × g  × g ( x)  x) 3. ( f /  f / g )( )( x  x)) = f  = f (( x)  x) if and only if  g   g ( x  x) ≠ 0  g ( x)  x) 4. ( f   f ◦ g )( )( x  x)) = f  = f (( x  x) ◦ g ( x)  x) = f  = f (( g   g ( x))  x))

Domain and Range Domain: all the possible values of  x  x Mathematical Impossibilities for Domain: o A fraction having a denominator deno minator of zero Any even numbered root of a negative number  Range: all the possible values of  y  y  

Even Functions  

 –  x A function f  function f is is even if  f (  f  ( –   x)) = f  = f (( x)  x) for all x all x in the domain of  f   f  The graph of an even function has symmetry across the y the y-axis -axis

Odd Functions  

 –  x A function f  function f is is odd if  f (  f ( –   x)) = –   f (( x)  f   x) for all x all x in the domain of  f   f  The graph of an odd o dd function has symmetry across the origin

Linear Functions 

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Linear functions are polynomials in which the largest exponent is 1 Lines that are closer to the horizontal have fractional slopes slopes o Lines that are closer to the vertical have slopes great er than 1 or less than -1 o The graph of a linear function is always a straight line Slope of a Line: m =  y2 –  y  y1  x2 –  x  x1 Slope-Intercept Form: y Form: y = mx + b Point-Slope Form: ( y  y –  y  y1) = m( x  x –  x  x1) Parallel lines have the same slope

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Perpendicular lines have slopes whose product is – 1 (negative reciprocals) Distance between two points: d = √ ( x  x2 –  x  x1)2 + ( y  y2 –  y  y1)2 The midpoint of a segment: M  segment: M = =  x1 + x2  y1 + y2 2 , 2 Perpendicular distance between a point and a line: d  d = = | Ax1 + By1 + C  C || 2 2 √ A + B The angle θ  between two lines, l 1 and l 2 : Tan θ  = m1 – m2 1 + m1m2 m1 is the slope of l  of l 1, and m2 is the slope of l  of l 2 o If tan θ  > 0, then θ  is the acute angle formed by the two lines o o If tan θ  < 0, then θ  is the obtuse angle formed by the two lines In order to find the other angle formed by the two lines, subtract the angle you o already have from 180° Whenever tan θ  = (a (a / 0), where a is an arbitrary constant, θ  = 90° o

Graphing Linear Inequalities 

If an inequality is “greater than or equal to” or “less than or equal to,” then the t he line is drawn as a solid line

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If an inequality is “greater than” or “less than,” then the line is drawn as a dotted line

Quadratic Functions   

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Quadratic functions are polynomials in which the largest expo nent is 2 The graph of a quadratic function is always a parabola 2 Quadratic Formula: x Formula: x = – b ± √b  – 4ac (roots of a quadratic function) funct ion) 2a 2 Discriminant: b  – 4ac 2 If  b  – 4ac > 0, then the equation has t wo distinct roots o 2 o If  b  – 4ac = 0, then the equation has a “double root” 2 If  b  – 4ac < 0, then the equation has no real roots o The sum of the two zeros (real or imaginary) of a quadratic function equals  – (b / a) The product of the two zeros (real or imaginary imaginary)) of a quadratic function equals (c / a)

Movement of a Function In relation to f  to f (( x):  x):  f (  f ( x)  x) + c is shifted upward c units  f (  f ( x)  x) – c is shifted shifted downward c units  f (  f ( x  x + c) is shifted to the left c units  f (  f ( x  x – c) is shifted to the right c units  – f ( f ( x)  x) is flipped upside down over the x the x-axis -axis  –  x)  f (  f ( –   x) is flipped left-right over the y the y-axis -axis │ f (  f ( x  x)│ is the result of flipping upward all of the parts of the graph that app ear   below the x the x-axis -axis       

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 f (│ x│) is the result of flipping to the right all of the parts of the graph that appear  to the left of the y the  y-axis -axis

Line Tests Vertical-Line Test: Any vertical line drawn can intersect a function only once. If a vertical line intersects 

a graph more than once, the graph isn’t a function

Horizontal-Line Test: Any horizontal line drawn can intersect a function with an inverse only once. If a horizontal line intersects a function more than once, t he domain of the function must  be limited in order for the function function to have an inverse 

Degrees of Functions  

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The degree of a po lynomial is the highest degree of any term in the po lynomial The degree determines at most how ho w many roots the polynomial will have These roots can be distinct or identical o An nth-degree function has a maximum of n of n roots and a maximum of (n (n – 1) extreme values in its graph

Inverse Functions  

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The inverse of a function f  function f is is denoted by f  by f  ( x)  x)  f (  f ( g   g ( x))  x)) = x = x The above statement states that f  that f (( x)  x) and g  and g ( x)  x) are inverse functions o -1 o The above statement can also a lso be written as g  as g ( x)  x) = f  = f  ( x)  x) -1 -1  f (  f ( x  x) ◦ f  ( x)  x) = f  = f  ( x  x) ◦ f (  f ( x)  x) = x = x -1  f  is not necessarily a function If  f (  f  (a) = b, then f  then f  -1(b) = a -1 The graph of  f   f  is the reflection of the graph of  f about  f  about the line y line y = x -1 Algebraically, the equation of  f   f  can be found by interchanging x interchanging x and y and y in the function f  function f and and solving for  y  y -1  f  can always be made a function by limiting the domain of  f  Switch the letter of variable restriction when taking the inverse o f  f   f 

Extra Tips 

Any graph that has symmetry across the x the x--axis isn’t a function because vertical-line test

it fails the