Trigonometric Derivative functions, Angles and Degree Measure, Solving Logarithmic and Exponential Equations, Properties of Logarithms, Graphing Logarithmic Functions, Exponential Functions, Inverse Functions, Composite Functions, Rational and Polyno...

The range of the inverse of f(x) = x³

( -∞ , ∞ )

(x+7) /(2x+1) > 2

(-1/2, 5/3)

(2-x) / [(x-5)(x+10)] ≤ 0

(-10, 2] ∪ (5, ∞)

(x+32)/(x+6)≤3

(-inf, -6]U[7, inf)

(x + 9) (x - 2) (x + 5) < 0

(-∞, -9) ∪ (-5, 2)

5x² + 20x + 30 > 3x² + 4x

(-∞,-5) ∪ (-3,∞)

-2

-10 + log₃ (x + 3) = -10

2

-10 + log₃ (x + 3) = -10

cotx

-csc^2x

cscx

-cscxcotx

cosx

-sinx

Steps to solve algebraically for the inverse of the function

1. Replace f(x) with y in the equation for f(x) 2. Interchange x and y 3. Solve for y. 4. Replace y with f⁻¹(x)

32

2log₄ x = 5

100

3log x = 6

6

3ⁿ⁻² = 81

1

4ⁿ⁺² = 64

The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as mc023-1.jpg, where mc023-2.jpg and is the least intense sound a human ear can hear. What is the approximate loudness of the dinner conversation, with a sound intensity of 10-7, Rajah has with his parents?

50 Db

minutes

60 of these equals 1 degree.

seconds

60 of these equals 1 minute

Exponent

A number placed above and to the right of another number to show that it has been raised to a powe

One-to-one Function

A property of functions where the same value for y is never paired with two different values of x (the function passes the horizontal line test)

terminal side

A ray of an angle in standard position that rotates about the center

degree

A unit used to measure distances around a circle. One degree equals 1/360 of a full circle.

Horizontal Line Test

A way to establish if a function is one-to-one when looking at a function's graph.

Vertical Line Test

A way to establish that a relation is a function.

Reflection about the line y = x.

A way to graphically see if two functions are inverses of each other.

standard position

An angle positioned so that its vertex is at the origin of a coordinate plane, and its initial side is the positive x-axis

What is mc017-1.jpg written as a single logarithm?

B) mc017-3.jpg

The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as mc026-1.jpg, where mc026-2.jpg and is the least intense sound a human ear can hear. Brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. The sound intensity level that Brandon can hear is 10-10. Ahmad, Brandon's neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is 10-4. How does Brandon's sound intensity level compare to Ahmad's?

Brandon's sound intensity is mc026-3.jpg the level of Ahmad's.

Which expression is equivalent to mc009-1.jpg?

C) mc009-4.jpg

Which expression is equivalent to mc014-1.jpg?

C) mc014-4.jpg

Vertex

Common endpoint of 2 rays

True! f[g(x)] is generally not equal to g[f(x)]. Consider f(x) = 2x, and g(x) = x - 3 f[g(x)] = 2(x - 3) = 2x - 6 g[f(x)] = (2x) - 3 = 2x - 3 f[g(x)] is not equal to g[f(x)].

Composition of functions is not commutative. True or false?

Which is the graph of a logarithmic function?

D

What is mc007-1.jpg rewritten using the power property?

D) mc007-5.jpg

reference angle

For an angle in standard position, the reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis.

Given two functions f[g(x)], step1: substitute the inner function g(x), for x step2: insert into outer function f(x) step3: perform operations step4: combine like terms

How do you compose two functions?

Step1: find the domain (restrictions) of the inner function Step2: combine the functions step3: find the domain (restrictions) of the composite function step4: compose domains.

How do you find the domain of a composite function?

The function y=log(x) is translated 1 unit right and 2 units down. Which is the graph of the translated function?

IT IS NOT C.

If you are given the graph of g(x)=log of 2x, how could you graph f(x)=log of 2x+5?

IT IS NOT Translate each point of the graph of g(x) 5 units left.

What is the range of y=log of 2(x-6)?

IT IS NOT all real number greater than 6

What is the range of y=log of 8x?

IT IS NOT all real numbers not equal to 0.

Which of the following is true about the base b of a logarithmic function?

IT IS NOT b<0 and b DOEST NOT EQUAL TO 1.

Which of the following is the inverse of y=3^x?

IT IS NOT y=log of 1/3x.

Inverse Function

If a function is named f, this can be written as f⁻¹

Step 1: Substitute g(x) for x f[g(x)] = f[x - 8]

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the first step?

Step 4: combine like terms = 3x² - 48x + 198.

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the fourth step?

Step 2: Insert into f(x) f[g(x)] = 3(x - 8)² + 6

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the second step?

Step 3: Factor = 3(x² - 16x + 64) + 6

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the third step?

Which expression is equivalent to log3(x + 4)?

NOT A Probably C

Given mc006-1.jpg and mc006-2.jpg, what is mc006-3.jpg?

NOT B

Given mc003-1.jpg and mc003-2.jpg, what is mc003-3.jpg?

NOT D

Which of the following is equivalent to mc002-1.jpg?

NOT D Probably A

(x+2)² < 0

No solution

Domain Restriction

Omitting specific values from a relation's set of input values, commonly to ensure that a function's inverse is also a function.

Which statement is true?

The graph of y=log of b(x) +4 is the graph of y=log of b (x) translated 4 units up.

initial side

The ray that is on the x axis when the angle is in standard position

Domain

The set of all input values of a relation.

Range

The set of all output values of a relation.

The domain of the inverse of f(x) = x²

[ 0 , ∞ )

(2x+5) / [(x+1)(x-1)] ≥ 0

[-5/2,-1) ∪ (1,∞)

True! these are two ways of writing the same thing.

[f o g](x) = f[g(x)] true or false?

Exponential Function

a function with a variable as the exponent

Horizontal translation

a transformation that moves a graph to the left or right

Vertical translation

a transformation that moves a graph up or down

Domain

all possible x-values

Range

all possible y-values

Starting value

also known as initial value; represents the y-intercept

quadrantal angle

an angle in standard position whose terminal side lines on the x or y axis

Exponential Decay

an exponential function that DECREASES from left to right

Exponential Growth

an exponential function that INCREASES from left to right

Horizontal Asymptote

an imaginary, horizontal line that a graph comes really close to, but does not cross

coterminal angles

angles that have the same initial and terminal sides

Sinx

cosx

if we choose to remove the component (x-4) as the inner function g(x), then we replace x for every g(x) in f. h(x) = (x-4)² becomes = f[g(x)], or = f(x-4) so f(x) = x² , and g(x) = x - 4 the answer depends on which component you choose to remove.

decompose h(x) = (x-4)²

What are the domain and range of f(x)=log(x=6)-4?

domain: x > 6; range: y > -4

step1: in g(x) = √(1-x), x ≤ 1 step2: in f[g(x)] = 3/(√(1-x) - 2) step3: in 3/(√(1-x) - 2), √(1-x) ≠ 2 , so (1-x) ≠ 4, finally x ≠ -3 step4: f[g(x)], Domain {x | x ≤ 1, and x ≠ -3 }

find the domains of f[g(x)], if f(x)= 3/(x-2) and g(x)=√(1-x)

step1: in f(x) = 4x − 6, x is all real numbers step2: in g(f(x)) = √f(x) = √(4x − 6) step3: in g(f(x)) = √(4x − 6) √(4x − 6) ≥ 0, so x ≥ (3/2) step4: in g(f(x)), Domain {x | x ≥ (3/2) }

find the domains of g[f(x)], if f(x) = 4x − 6, and g(x) = √x .

when you break into simpler functions from a more complicated function; finding the components of a function.

function decomposition

The inverse of f(x) = x + 11

f⁻¹(x) = x - 11

The inverse of f(x) = x³ + 1

f⁻¹(x) = ³√(x-1)

The inverse of f(x) = (x + 1)³

f⁻¹(x) = ³√x - 1

The inverse of f(x) = 2(x - 16)

f⁻¹(x) = ½x + 16

The inverse of f(x) = 2x - 16

f⁻¹(x) = ½x + 8

Rate of change

how fast or slow a graph is changing; also known as slope for linear functions

7

ln(2x-3) = ln 11

0

log (10 - 4x) = log (10 - 3x)

3

log 5x = log(2x + 9)

96

log x - log 6 = 2 log 4

4

log₂ x + log₂(x - 3) = 2

-3

log₂(x - 5) - log₂(x - 2) = 3

64

log₄ x = 3

Percent

parts per 100

tanx

sec^2x

secx

secxtanx

Base

the number or variable being raised to a power

X-intercept

the point where a graph crosses or touches the x-axis

Y-intercept

the point where a graph crosses or touches the y-axis

Evaluate

when you "plug in" a number or variable

Which of the following is a logarithmic function?

y=log of 3x