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## What are the domain and range of f(x) = 2(3x)?

Option A. Domain = (-Infinity, Infinity); Range (0, Infinity)

Solution:

f(x)=2^(3x)

a) This is an exponential function, and we don’t have restrictions for the independent variable “x” in the exponent, then the Domain of f(x) is all the real numbers:

Domain f(x) = ( – Infinity, Infinity)

b) To find the range we can find the inverse function f^(-1) (x). The domain of the inverse function is the range of the original function f(x):

y=f(x)

y=2^(3x)

Isolating x: Applying log both sides of the equation:

log y = log 2^(3x)

Applying property of logarithm: log a^b = b log a; with a=2 and b=3x

log y = 3x log 2

Dividing both sides by 3 log 2:

log y / (3 log 2)=3x log 2 / (3 log 2)

log y / (3 log 2)=x

x=log y / (3 log 2)

Changing x by f^(-1) (x) and y by x:

f^(-1) (x) = log x / (3 log 2)

This is a logaritmic function and the argument of the logarithm must be greater than zero, then the Domain of the inverse function is:

x>0→Domain f^(-1) (x) = (0, Infinity)

The domain of the inverse is the range of the original function:

Range f(x) = Domain f^(-1) (x)

Range f(x) = (0, Infinity)

Domain f(x) = ( – Infinity, Infinity)

Range f(x) = ( 0, Infinity)

How do you find the domain and range of f(x)=2/(3x-1 – How do you find the domain and range of # f(x)=2/(3x-1)#? Algebra Expressions, Equations, and Functions Domain and Range of a Function. 1 Answer Narad T. May 26, 2018 The domain is #x in (-oo, 1/3) uu(1/3,+oo)#. The range is #y in (-oo,0) uu(0,+oo)# Explanation: The denominator must be #!=0# Therefore,to find the domain and range of the function: f(x) = \frac{x}{2 – 3x} Domain: for domain check the denominator of the function,and find those values of x for which function has finite value,thus for any rational number to be finite its denominator should not equal to zero. here 2-3x=0. 2=3x. x=2/3. So for x= 2/3 the function f(x) has to beDomain and range » Tips for entering queries. Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for the domain and range. domain of log(x) (x^2+1)/(x^2-1) domain; find the domain of 1/(e^(1/x)-1) function domain: square root of cos(x)

Find the domain and range of the function: f(x) = [tex – Answer to find the domain and range for the given function: f(x) =3x-2 Domain = (-3,0,3) Range = ?…What are the domain and range of the real-valued function f(x)=2/3x? A. The domain is all real number except 0. The range is all real numbers except 0. B. The domain and the range are all real numbers. C. The domain is x>0. The range is f(x)>0. D. The domain is all real numbers except 0. The range is all real numbersPrecalculus Find the Domain and Range f(x)=x^3-3x+2 The domainof the expressionis all real numbers except where the expressionis undefined. In this case, there is no real number that makes the expressionundefined.

Domain and Range Calculator: Wolfram|Alpha – f(x) = 3x + 5 => y = 3x + 5, here it is a straight line increasing with slope 3 and y-intercept 5,so both domain and range of the given function is all real number (-∞,∞) 13 views Related AnswerTranscript. Example 21 Find the domain of the function "f" (x) = (" " 2 + 3 + 5)/( 2 5 + 4) "f" (x) = (" " x2 + 3x + 5)/(x2 5x + 4) = (x2 + 3x + 5)/(x2 4x + 4) = (x2 + 3x +5)/( (x 4) 1( 4)) = (x2 + 3x + 5)/(( 4)(x 1) ) In real numbers , the denominator cannot be zero Hence (x 4) (x 1) 0 x 4 and x 1 Hence the domain of the function will be all real numbers except 1 and 4 Hence, the domain = RThe range is either y>=-5" "or" "y<=-5. Since a is positive, the parabola opens upwards. So the range has to be y is greater than or equal to something. So the range is y>=-5. Domain and range are easier to find when you have the graph of the equation:graph{3x^2 – 5 [-10.59, 9.41, -6.16, 3.84]}

**How to determine, domain range, and the asymptote for an exponential graph** – Now, what I'd like

to do is at least go into some definitions or

at least look into the graph.

The basic parent graph of our

exponential is fairly simple. All right. We do f of x so you could

say, here's your f of x axis. Here's your x-axis. There's a couple

important points. One is for all

exponential graphs, they're all going to have

the exact same y-intercept unless there's a transformation. But the parent graph, it does

not matter what the base is, y equals or f of x equals. It doesn't matter,

ladies and gentlemen, if it's f of x equals 8 to the x

or if f of x equals 2 to the x. Either one of these

graphs, these bases are not going to affect

our transformation. These are going to be

more of the dilation on how the graph is

going to grow or descend. But as far as what you

need to understand, the reason why they are going

to have the exact same– and we'll talk about how

to do it with a table– is because if I take

any of these, remember, to find the y-intercept,

you set x equal to 0. When I say, hey, find the

x-intercept, you put 0 in for– or find the y-intercept,

you put 0 in for x. Well, what you

guys should notice is it doesn't matter

what the base is. Your y-intercept's always

going to be 1, right? Yes. OK. So a couple things

I want to talk about with this exponential

graph– and mainly, I want to really get into the

domain and range and then also talk about the asymptote. Now, I know some of

you, we very briefly talked about rational functions. Well, we didn't talk

about it this year, but in previous classes, we

talked about rational functions and asymptotes. Remember, an asymptote is

a line that your function is going to approach. And generally, you could

say that they usually do not cross your asymptote. They're just approaching that. So this graph, the way that

I've graphed it, looks like– and behavior– looks

like it kind of goes up and to the right and

then down to the left. But it looks like it's tapering

off when it goes to the left, that it's not going to get

into the negative quadrant. And that is true. And let's look at this

and see the reason why. So I'm just going to pretend– let's do f of x equals–

let's just do 2 to the x. We'll keep it simple. So if I was going to

create a table for this, I would do x and f of

x values and I'm just going to pick values

that I want to pick. Now, ladies and

gentlemen, you don't need to pick crazy numbers. Just pick some very

simple numbers. So what I want to

explain to you guys is let's just pick negative

2, negative 1, 0, 1, and 2. Now, remember, when

graphing, by using a table, we just take those values

and we plug them in. So this is 2 to the

negative second power. This'd be 2 to the

negative first, 2 to the 0, 2 to the first, and 2

to the second power. Now, a couple of things I

want you guys to understand– 2 to the negative

second power, we have to be very

careful with this. A lot of people say it's

going to be a negative number. And remember, when we

look at this graph, if I said 2 to the negative

second power, that means it'd be down to negative 4. That doesn't make sense. So we need to remember what are

our properties of exponents. And our properties

of exponents state that if you have x raised

to a negative power, that is equivalent to

1 over x to that power. That's the properties

of exponents that we went back and reviewed. So it's very important

that you guys understand that because this is 1 over 2

squared, which is equal to 1/4. This is 1 over 2 to the

first power, which is 1/2. 2 to the 0 power is just

going to equal 1, 2, and 4. So what I want you

guys to understand is if I keep on getting larger

and larger negative numbers, is this ever going to get to 0? No. No. It's just going to keep on

being a bigger fraction. If you do negative

3, so then you'll have 2 to the

negative third power. Well, that's just equal to 1/8. What about if you do negative 4? Well that's going to be 2 to

the negative fourth power, which equals 1/16. And you're going to keep on

getting a larger and larger and larger and larger number. But it's never going to go

to 0 because you're always going to have a number 1

divided by another number that's going to keep on getting

larger and larger. So therefore, ladies

and gentlemen, you can see that as

we go to the left, as we keep on getting negative

and negative and negative, this is never going to get to 0. It's going to keep

on approaching 0. If you keep on plugging

this in, do 1/8. Do 1/16. Do 1/32. Do 1/64. That number keeps on

getting smaller and smaller and approaching 0 but it's

never ever going to approach 0. Therefore, when we're

talking about that, we can say that the

asymptote is y equals 0. So you guys are going to want to

make sure you write that down. Your asymptote is y equals 0. And now that we

have an asymptote, we can also start talking

about domain and range. So if I look at the domain of an

exponential function, remember, the domain is going to be

the set of all x-values that are going to make

your function true. So what that means is if

I choose an x-value here, is there a point on the graph? Yes. If I choose an x-value here,

is there a point on the graph? Yes. So for each one

of those points, I have a value that's going to

make it true for that function. What about if I pick

an x-value over here? Is it going to be on the graph? Yeah. And it will because this

graph continues but it's going to be way up there. And if I choose a value

all the way over here, it's still going

to be on the graph but I just obviously

can't represent it because this graph– here's another thing. You guys can tell

this is a function. Remember, what we talked

about polynomials is it has to be continuous. This is a continuous graph. There are no breaks

or sharp turns and the graph

continues infinitely in the negative direction

and in my positive direction. So therefore, the domain is from

negative infinity to infinity. And that's going to be true

for all of your logarithmic– or I'm sorry– your

exponential functions. Every x-value you plug

in there– and let's put a case in there. Let's look at our function

f equals 2 to the x. Is there a number that you can

think of that you cannot plug in for x that would make

this equation not true? Is there a number you

can't plug in for x– obviously, the real

number system because x is any real number. But think about any real number. Can you put in 2

to the 99 million? Yes. Yeah. Can you put 2 to the

negative 1 million? Yeah. Yeah. Every number, you

can plug in for x. So the domain is

all real numbers. You can plug in

any number you want to between negative

infinity and infinity and you will get one

single output value. But let's talk about the range. The range, remember,

is the output. So whatever number

you plug in for x, are you able to get

any possible number? Now, I showed you

guys when you keep on getting larger you're

just going to keep on getting higher and higher numbers. And when you get to

smaller numbers, negatives, you keep on getting

smaller and smaller numbers but you're never

going to get to 0. So our range– is it possible

for me to put a number in for x and to get a negative number? Can I put two

raised to some power and get a negative number? No. No, because when I put

negative numbers in for there, it just makes it a fraction. And when I put positive

numbers, it just makes it a larger number. So it's impossible for me

to put a value in for x and to get a negative number. It's also impossible for

me to put a value into x and get the number 0. because when I put

0 into x, I get 1. So our range does

not include 0– I'm sorry– but it goes

from negative infinity to 0. And we're going

to use parentheses because 0 and negative infinity

are not included in the answer. So that's going

to be your range. So that's when we talked

about asymptotes and range for your exponential functions. Cool? All right. OK. You guys don't seem too– .

**g(x)=sqrt(16-x^2) Find domain, range, increasing/decreasing intervals** – E r .

**How to determine the domain and range and graph** – .