Download Brief Notes On Functions

General Function Information:
1. Function: A function is a rule that associates an element of one set, called the domain uniquely
with an element of another set called the range.
2. Inverse Functions: f (x) and g(x) are inverse functions if f (g(x)) = x and g(f (x)) = x. We
normally write g(x) = f −1 (x). The graph of the inverse of a function is the reflection in the
line y = x of the graph of the function.
3. Even and Odd Functions:
• A function is even if f (−x) = f (x), its graph is symmetric about the y-axis.
• A function is odd if f (−x) = −f (x), its graph is symmetric about the origin.
4. Piecewise Defined Functions: For Example


x2 + 1
x≤0
−3x + 1 0 < x ≤ 4
f (x) =

−x3 + 20x
x>4
5. Algebraic Functions:
• Linear Function:
∆y
– Slope: m = rise
run = ∆ x
– Point Slope Form: (y − y0 ) = m (x − x0 ) or y = m (x − x0 ) + y0
– Slope Intercept Form: y = m x + b
• Power Function: y = xr , where r ∈ R
• Polynomial: y = an xn + an−1 xn−1 + · · · + a1 x + a0 , where ai is any number, an 6= 0, and
all the exponents are non-negative integers. If an = 1 the polynomial is called monic.
– The degree of a polynomial is the highest power of the variable.
Even Degree Polynomial
Odd Degree Polynomial
– A root or x-intercept of a polynomial are the places it crosses the x − axis which is
the same as the values of x for which the polynomial equals 0.
– The number of real roots of a polynomial is less than or equal to its degree.
– The y-intercept of a polynomial is where it crosses the y-axis and is the constant
term of the polynomial.
p(x)
• Rational Functions: y = q(x) where p(x) and q(x) are polynomials.
• General Algebraic Function: Any combination of x’s involving addition, subtraction,
multiplication, division, and raising to fixed powers.
6. Transcendental Functions:
• Trigonometric Functions: sin(x), cos(x), tan(x), sec(x), csc(x), and cot(x).
sin(x)
cos(x)
tan(x)
csc(x)
sec(x)
cot(x)
• Exponential Functions: y = a bx where a and b are real numbers, and b, the base, is
greater than 0.
bx , b > 1
bx , 0 < b < 1
• Logarithmic Functions: y = logb (x) is the inverse function of y = bx , and b, the base, is
a real number greater than 0.
logb (x), b > 1
• Common Log: y = log(x), the logarithm base 10.
• Natural Log: y = ln(x), the logarithm base e.
7. Parametric Functions: A parametric function or parametrized curve is one in which each
variable or coordinate such as x or y is a function of a third variable. The third variable is
frequently called t for time. For example the straight line
y = 34 x + 9
could be written parametrically as
hx(t), y(t)i = h4t, 3t + 9i
both of which will produce the same graph for different
values of t, x, or y.
Note that because parametric functions are written with a different function for each coordinate they can represent the same function in more than one way. For example the above line
can also be written
hx(t), y(t)i = h8t, 6t + 9i
or
hx(t), y(t)i = h2t + 4, 1.5t + 12i
so that we can with parametric equations represent speed and motion in a way we can not
with regular functions. They can also represent curves that can not be represented by our
usual functions. For example
hx(t), y(t)i = t3 − t, t2 − 1
is a curve that does not pass the vertical line test and so is not a function in the regular since.
8. Implicit Functions: An implicit function is a function in which the variables, x and y, depend
upon each other in some way but frequently you can not simply plug in a value for one variable
and get the value for the other. For example in the equation
x2 + 4x + 4 + y 2 − 6y + 9 = 29
y and x are implicitly functions of one another. However if we let x = 0 then we get the
equation
y 2 − 6y + 13 = 29
which has solutions y = −2 or y = 8. As with parametric curves implicit functions can
represent curves that are not functions in the traditional since such as circles (which the
equation above represents), ellipses, and hyperbolas.
Quadratics, Polynomials, and Rational Functions:
• Quadratics: A quadratic polynomial is a polynomial of the form
y = ax2 + bx + c or y = a(x − r)(x − s), a 6= 0
• Roots: A root or x-intercept of a polynomial are the places it crosses the x − axis which is the
same as the values of x for which the polynomial equals 0.
– For a quadratic written as y = a(x − r)(x − s) the roots are x = r and x = s.
– If the quadratic is monic, y = x2 + bx + c, then the product of the roots equals c and
their sum is −b. That is if
x2 + bx + c = (x − r)(x − s)
then −b = (r + s) and c = (r · s).
– In general the roots of a quadratic can always be found using the quadratic formula:
√
−b± b2 −4ac
x=
2a
• Polynomial Long Division: Examples
2
x + 5x + 8
x − 2 x3 + 3x2 − 2x − 16
x3 − 2x2
5x2 − 2x − 16
5x2 − 10x
8x − 16
8x − 16
0
x3 − 9x2 + 17x − 22
x2 + 2x + 1 x5 − 7x4
+ 3x2 − 9
x5 + 2x4 + x3
−9x4 − x3 + 3x2 − 9
−9x4 − 18x3 − 9x2
17x3 + 12x2 − 9
17x3 + 34x2 + 17x
−22x2 − 17x − 9
−22x2 − 44x − 22
27x + 13
• Factors and Factoring: The factors of a polynomial are the other polynomials that divide
evenly into it. For example from above y = x − 2 is a factor of y = x3 + 3x2 − 2x − 16. If
x = r is a root of a polynomial then y = x − r is a factor and in general can be factored out
using polynomial long division.
• Graphs of Polynomials: As x goes to positive and negative infinity all even degree polynomials
look similar to y = x2 and all odd degree polynomials look similar y = x3 .
Even Degree Polynomial
Odd Degree Polynomial
• End Behavior: The end behavior of a rational function is what the what happens to the graph
of or output values of the polynomial as the value of the input x goes to positive or negative
infinity.
– You find the end behavior of a rational function by dividing the numerator by the
denominator and discarding the remainder. For example
from above we know:
27x+13
x5 −7x4 +3x2 −9
= x3 − 9x2 + 17x − 22 + 2
x2 +2x+1
x +2x+1
and so the end behavior is y = x3 − 9x2 + 17x − 22.
p(x)
– Horizontal Asymptote: Given y = q(x) then y has a horizontal asymptote if the degree of
the function in the denominator is greater than or equal to the degree in the numerator. If
the degree in the denominator is greater the asymptote is y = 0 otherwise it is equal to the
leading coefficient of the numerator divided by the leading coefficient of the denominator.
– Oblique Asymptote: An oblique or slant asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator so that the end behavior
of the function is a line.
• Vertical Asymptotes: A vertical asymptote occurs at values of x where the denominator is equal
to zero and the numerator is not. Points, values of x, where the numerator and denominator
are both zero prior to simplification and where the denominator is not zero after simplification
are called points of discontinuity.
Trigonometry to Know or Recognize
Definitions:
sin(x)
cos(x)
tan(x)
csc(x)
opposite
hypotenuse θ
opp.
y
1. sin(θ) = hyp. = r
adj.
2. cos(θ) = hyp. = xr
sin(θ)
opp.
y
3. tan(θ) = adj. = x = cos(θ)
adjacent
hyp.
1
4. sec(θ) = cos(θ)
= adj. = xr
hyp.
1
5. csc(θ) = sin(θ)
= opp. = yr
sec(x)
(x, y)
cot(x)
Z
rZ
Zθ
r
adj.
1
= opp. = xy
6. cot(θ) = tan(θ)
Basic Identities:
7. cos(2θ) = cos2 (θ) − sin2 (θ)
Pythagorean Identities:
1. sin2 (θ) + cos2 (θ) = 1
8. cos(2θ) = 1 − 2 sin2 (θ)
2. 1 + cot2 (θ) = csc2 (θ)
9. cos(2θ) = 2 cos2 (θ) − 1
3. tan2 (θ) + 1 = sec2 (θ)
General Stuff:
Sum & Difference Rules:
4. sin(A±B) = sin(A) cos(B)±cos(A) sin(B)
5. cos(A±B) = cos(A) cos(B)∓sin(A) sin(B)
10. sin(θ) = cos(90◦ − θ)
11. cos(θ) = sin(90◦ − θ)
12. sin(−θ) = − sin(θ)
Double Angle Rules:
13. cos(−θ) = cos(θ)
6. sin(2θ) = 2 sin(θ) cos(θ)
Angles to Know:
θ
0◦
sin(θ)
0
cos(θ)
1
tan(θ)
0
sec(θ)
1
csc(θ)
cot(θ)
30◦
1
2
√
3
2
√
3
3
√2
3
undef.
60◦
√
√
2
2
√
2
2
1
√
√
undef. 2
√
45◦
3
1
3
2
1
2
√
3
90◦
180◦
270◦
1
0
−1
0
−1
0
undef. 0
undef.
undef.
2
2
undef. −1
2
√2
3
√
3
3
1
30◦
undef. −1
1
undef. 0
√
3
2
◦
60
1
2
1
45◦
√
0
2
2
√
2
2
Logarithmic and Exponential Functions:
In general an means we take a product of n copies of a
n
n
z
}|
{
a = a · a · a···a
when n is an integer. This idea can be extended to all positive and negative exponents.
The log base b of a number x is the number y such that by = x and we write
y = logb (x) or x = by
Note that we assume b is greater than 0.
Rules for Exponents:
Rules for Logs:
• an am = am+n
• logb (x) is that number y such that by = x
n
• aam = an−m
• log(x) = log10 (x)
• (an )m = anm
• ln(x) = loge (x)
• an bn = (ab)n
n
n
• abn = ab
• logb (b) = 1 because b1 = b
• logb (1) = 0 because b0 = 1
• a−n = a1n , a 6= 0
√
• a1/n = n a
• logb (M N ) = logb (M ) + logb (N )
• logb M
N = logb (M ) − logb (N )
• a1 = a
• logb (M p ) = p log(M )
• logb N1 = − logb (N )
• a1/2 =
• a1/3
√
a
√
= 3a
• a0 = 1, a 6= 0
• a−1 = a1 , a 6= 0
• logb (bx ) = x
• blogb (x) = x
log (M )
• Change of base formula logb (M ) = logc (b)
c
Graphs:
bx , b > 1
bx , 0 < b < 1
logb (x), b > 1