Math 118 Exam 1 Study Guide

Area of a circle

A\=πr²

Area of a rectangle

A\=lw

Area of a triangle

A\=1/2bh

Volume of a cylinder

V\=πr²h

Volume of a rectangular solid

V\=lwh

Volume of a triangular prism

V\=(1/2bh)h

Circumference of a circle

C\=2πr

Pythagorean theorem

a^2+b^2\=c^2

Even symmetry: if \_____, then \_____

if (a,b), then (-a, b)

Odd symmetry: If \_____, then \_____

if (a,b), then (-a,-b)

Symmetric about the line y\=x: If \_____, then \_____

If (a,b), then (b,a)

Describe the symmetry between f(x) and the inverse of f(x)

symmetric about the line y\=x

When evaluating a problem and you do not find an answer then it is

undefined

When solving a problem and you do not find an answer then it is

no solution

Is it okay for multiple inputs to give the same output?

yes

Is it okay for one input to give multiple outputs?

no

What are the domain restrictions for a log function?

x \> 0

What is the criteria for functions to be identical/equivalent?

they must have the same domain and ALL the same points

List the asymptotes for y\=2^x

horizontal

List the asymptotes for y\=1/x

horizontal and vertical

List the asymptotes for a log function

vertical

What is a correspondence that *maps* EACH element in x to a UNIQUE element in y?

function

Name the other terms for input (there's 4)

x, domain, independent variable, argument

Name the other terms for output (there's four)

y, range, dependent variable, value

What is evaluating?

finding an output for the given input (at most one answer)

What is solving?

finding all inputs for a given output (infinite answers)

What is finding?

either evaluating, solving, or finding an expression

What is an increasing function?

What is a decreasing function?

\

What is a concave up function?

\

What is a concave down function?

\

If a function has even symmetry then f(-x) \=

f(x)

If a function has odd symmetry then f(-x) \=

-f(x)

What are three domain restrictions?

can't divide by zero, can't take the square root of a negative number, can't take the log of a negative number or zero

When do you need a sign chart?

if multiple linear factors are in the same even root or the same log

What is an equation for the thing you want to maximize or minimize called?

objective function

If the objective function has more than one input variable, you must find any equations that relate to your input variables. What are these called?

constraint equations

General equation for a vertical shift

f(x) + c

How do you demonstrate a vertical shift on a set of points?

Either add or subtract c from the y-values

General equation for a vertical stretch/compression

cf(x)

When is cf(x) a vertical stretch?

when c is greater than one

When is cf(x) a vertical compression?

when c is greater than zero, but less than one

How do you demonstrate a vertical stretch on a set of points?

multiply y values by c

How do you demonstrate a vertical compression on a set of points?

divide y values by c

General equation for horizontal shift

f(x+c)

How do you demonstrate a horizontal shift on a set of points?

Either add or subtract c from the x-values

If c is positive during a horizontal shift, do you add or subtract c from the x-values?

subtract

If c is negative during a horizontal shift, do you add or subtract c from the x-values?

add

General equation for a horizontal compression/stretch

f(cx)

When is f(cx) a compression?

when c is greater than one

When is f(cx) a stretch?

when c is greater than one but less than zero

How do you demonstrate a horizontal stretch on a set of points?

multiply x-values by the reciprocal of c

How do you demonstrate a horizontal compression on a set of points?

divide x-values by c

What is -f(x)?

vertical flip, reflection about the x-axis

How do the points of a function change when there is a vertical flip/reflection about the x-axis?

x remains unchanged, y switches signs

What is f(-x)?

horizontal flip, reflection about the y-axis

How do the points of a function change when there is horizontal flip/reflection about the y-axis?

x switches signs, y remains unchanged

What must you do if you have a horizontal shift and another horizontal transformation?

factor the argument to see the true shift

Steps for a sequence of transformation

horizontal shift -\> stretches/compressions -\> reflections -\> vertical shifts

When should you plug points into the equation to check you did the transformation correctly?

always

\___________ is the set of numbers in the domain in the domain of a resulting composite function AND the domain of the inner function.

domain of a composite function

In order to ensure a composite function remains a function, you need \_____

domain restrictions

How do you find inverses algebraically?

switch x and y and solve for y

The input of f is the \_____ of f inverse.

output

The input of f inverse is the \_____ of f.

output

If the point (a,b) is on f, then the point \_____ is on f inverse.

(b,a)

Are the domain and range of f and f inverse the same?

no, they switch

f^1(x) is the \_____. (f(x))^1 is the \______.

inverse, reciprocal

What do you do if a function is not one-to-one?

restrict it's domain

Draw y = |x|

Draw y=x^2

Draw y=x^3

Draw y=√x

Draw y=2^x

Draw y=1/x

Draw y=sin(x)