Chapter 1 Flashcards
Definition of Function
A function”f” in a rule that assins to each input “x” in a set “a”, has exactly one output y=f(x) in the “b” set
4 Ways to represent functions
Verbally, Numerically, Visually, and Algebraically
What is the VLT (Vertical Line Test) for?
Test if it is a function
What is the HLT (Horizontal Line Test) for?
Test for 1-1 ness
Definition of Even Function
If a function “f” satisfies f(-x) = f(x) for all elemental (triangle thing –> x –> E –> D) then “f” is called an even function
Definition of Odd Function
If function “f” satisfies f(-x) = -f(x) triangle thing –> x –> E –> D then “f” is called an odd function
It is not a polynomial function if there are…
Absolute values, variable in the denominator, etc
Which way does a graph open?
Positive = Up
Negative = Down
Vertical Shifting
y = f(x) –> y = f(x) + c
+c = up
-c = down
Horizontal Shifting
f(x) = x^2
g(x) = (x+1)^2 –> left
h(x) = (x-2)^2 –> right
Reflections with respect to the y-axis
y = f(x) and y = f(-x)
Reflections with respect to the axis
y = f(x) and y = -f(x)
Definition of a function
Suppose “f” is a function defined on some open interval that contains “a”, except possibly at “a” itself. Then lim (x–>a) f(x) –> L if as “x” approaches “a” (on either side of “a”). f(x) is close to L.
In short, as x–>a, f(x) –> L
Definition of one-sided limits
Lim (x–>a-) f(x) = L if as “x” approaches “a” from the left f(x) approaches L
Lim (x –> a+) f(x) = L if as “x” approaches “a” from the right f(x) approaches L
Definition of a limit #1
Let “f” be a function defined on some open interval that contains the number “f” expect possible at itself. Then lim f(x) = if triangle thing epsilon is greater than 0, epsilon delta is less than 0 such that if 0 < l x-a l < epsilon then l f(x) - L l < epsilon
Definition of a limit #2
Limit from the right; Let f(x) be defined on an open interval (a,b). Then lim (x–>a+) = L, if triangle thing epsilon > 0, backwards E (for all) delta > 0 such that if 0 < x-a < delta the l f(x) - L l < E.
Limit from the left: Let f(x) be defined on an open interval (c,a). Then lim (x –> a-) f(x) = L if triangle thing E > 0, backwards E delta > 0 such that if negative delta < x-a < 0 then l f(x) - L l < E
Limit Laws
Suppose that “c” is a constant and the limits lim (x–> a) f(x) and lim (x–>a) g(x) exist. Then …
Limit Law Theorem
If “f” is a polynomial then lim (x–>a) f(x) = f(a)
If “f” is a rational function and “a” is in the denominator of “f” then lim (x–>a) f(x) = f(a)
If f(x) < ___ g(x) where “x” is near “a” (except possibly at “a”) and the limits of “f” + “g” both exist as “x” approached a, then lim (x–>a) f(x) < __ lim (x–>a) g(x)
Squeeze Theorem
If f(x) < __ g(x) < ___ h(x) where “x” is near “a” (except possibly at “a”) and lim (x–>a) f(x) = lim (x–>a) h(x) = L then lim (x–>a) g(x) = L
Definition of continuity
A function “f” is continuous at “a” if lim (x–>a) f(x) = f(a)
Notice that the definition implicitly requires
(IF 1 + 2 works then 3 works too)
1. f(a) is defined as long every integer is defined
2. lim (x–>a) f(x) exists
3. lim (x–>a) f(x) = f(a)
Definition of one-sided continuity
A function “f” is continuous from the right at “a” if lim (x–> a+) f(x) = f(a) + “f” is continuous from the left at “a” if lim (x–>a-) f(x) = f(a)
A function “f” is continuous on an interval “f” if it is continuous at every number in the interval
One-sided continuity Theorems
If “f” + “g” are continuous at “a” and “c” is a constant, then the following are also continuous at “a”
- Any polynomial is continuous everywhere, ie, everywhere
- Any rational function is continuous it is defined
- The following types of functions are continuous at every number in their domains: polynomials, rational functions, exponential functions, logarithmic functions, root functions, + trigonometric functions [And an arithmetic combination of these functions]
- If “g” is continuous at “a” + “f” is continuous at g(a), then (fog)(x)=f(g(x)) is continuous at “a”
- If “f” is continuous at “b” and lim (x–>a) g(x) = b , then lim (x–>a) f(g(x)) = f(b)
Intermediate value Theorem
Suppose that “f” is continuous on [a,b] and let N be any number b/n f(a)+ f(b) when f(a) = f (b). Then backwards E c epsilon (there exists at least one “c” epislon) (a,b) such that f(c) = N
