Ch. 1: Functions and Models
f(x)±g(x)=
(f ±g)(x) -domain: A ∩ B
f(x)*g(x)=
(f*g)(x) -domain: A ∩ B
f(x)/g(x)=
(f/g)(x) -domain: {x ∈ A ∩ B | g(x) ≠ 0} (We can use x-values that are in both A and B as long as they don't make g(x)=0)
If y=f(x) changes to y=f(ax)...
-"a" changes domain -if a section of the graph was 1 unit wide and y units tall, it will become 1/a units wide and stay same height -if a<0 graph reflects over y-axis
If y=f(x) changes to y=af(x)...
-"a" changes range -we get a vertical stretch/compression by a factor of a -if a<0, graph flips upside down (over x-axis)
To find domain look for:
-square roots (what's inside must be greater than or equal to zero) -logs (can't take the log of a negative number or zero) -fractions with x in the denominator (denominator can't be zero)
What are the three types of calculus questions?
1. Area problem 2. Tangent problem 3. Limit of a sequence
What are the 4 ways to describe a function?
1. Verbally 2. Numerically (table) 3. Visually (graph) 4. Algebraically (formula)
Function
A rule that assigns each x to one y (and only one y)
Independent variable
A variable (often denoted by x ) whose variation does not depend on that of another
Dependent variable
A variable (often denoted by y ) whose value depends on that of another
_____________________________ is a piecewise function
Absolute value
Range
All possible y values of a function
Domain
All the x values we are allowed to plug into a function
If y=f(x) changes to y=|f(x)|...
Anything below x-axis flips up (all y values become positive)
What are the two types symmetry that the graphs of functions can have?
Even and odd
Odd function
Graph is symmetrical with respect to the origin (reflect it once over x and once over y); f(-x)=-f(x)
Even function
Graph is symmetrical with respect to the y-axis; f(x) = f(-x)
If y=f(x) changes to y=f(x+c)...
Graph moves left/right c units
If y=f(x) changes to y=f(x)+c...
Graph moves up/down c units
If y=f(x) changes to y=f(|x|)...
Left half of graph disappears and we get a mirror image of the right half
To find range...
Look at graph and figure out what parent function looks like
How to tell algebraically if a function is odd, even, or neither:
Plug in (-x): is the result... -the same (even) -exactly opposite (odd) -different somehow (neither)
Inverse functions are...
Reflections of each other over the line y=x
Degree of a polynomial
The highest exponent in a polynomial (# of times graph can cross x-axis)
Combinations of functions
The sum, difference, product, or quotient of functions *For further reference, consider 2 functions, f(x) and g(x) -domain of f(x): A -domain of g(x): B
How to tell if a rule is a function:
Vertical line test (if it touches more than one spot it fails)
Composition of functions
What happens when we put one function inside another function?
Piecewise function
a function composed of 2 or more functions
Asymptote
a line that a graph approaches but never crosses
(f ∘ g)(x)=
f(g(x)) -to find domain, look at both original and final functions
A function is increasing on an interval, I, if:
f(x1)<f(x2) whenever x1<x2 in I
A function is decreasing on an interval, I, if:
f(x1)>f(x2) whenever x1<x2 in I
What are examples of even functions?
y=x^2; y=cos x
What are examples of odd functions?
y=x^3; y=sin x
