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

source : e-eduanswers.com

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 – 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.

Find the domain and range of the function: f(x) = [tex

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]}

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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 – .