in-3-11-each-graph-shows-a-polynomial-functions-from-the-set-of-real-numbers-to-the-set-of-real-numb

Question

In $3-11$ , each graph shows a polynomial functions from the set of real numbers to the set of real numbers.
a. Find the real roots or zeros of the function from the graph, if they exist.
b. What is the range of the function?
c. Is the function onto?
d. Is the function one-to-one?
(GRAPH NOT COPY)

Finian Lickona · Accepted Answer

Okay, So for this exercise were given the graph of a function and we are asked to determine ah, what the roots of the function are. Ah, what the range of the function is and whether the function is onto or end or 1 to 1. So the first thing I&#x27;m gonna do, I&#x27;m going to actually right out of the equation for this graph. This is simply the graph of the function Y equals two X plus two. And if we look at the graph right here, we see very clearly that, uh, X equals negative. One is a root or intercept of this particular function. So the root is the only route is negative one because it&#x27;s a linear function. And now I need to determine the range of the function, which is all the possible output values it could have. Now, since this is a linear function, it extends from negative infinity to positive infinity in both directions. So the range is all real values, so range all real numbers, and we can go ahead and use algebraic notation. Big fancy are to denote all real numbers. So now we need to determine if the function is onto and a function is on two. If given every output, there is some input that produces that output. So we can clearly see that for any why value that we want Thio find on this line There is some X value that will, when we plug it into our equation generate that Why value So the function is aren&#x27;t too. Now we need to determine if the function is 1 to 1. A 1 to 1 function is a function where for ever every output there is on Lee one input that produces that output. So we can see that if we wanted to produce an output of positive too. The only input we can give is zero. Because if we plug zero in the ER, no other number is going to generate positive too. Except for zero on this process can be repeated for any potential input of potential output pairs. So the answer is yes. This is a 1 to 1 function. Generally, in fact, no, consistently all linear functions are going to be both one toe

Finian Lickona · Answer

Okay, So for this exercise, we are given this graph and we are asked to find the real roots the range and determine if the function is on to end or one the one. So I&#x27;m gonna start by finding the roots which in this case, is very easy. We just have to look at where the graph intersects the X axis that clearly occurs at X equals negative one and X equals three. So the roots, our X equals negative one and X equals negative three. And now we need to determine the range, which is all of the possible output values for the function, essentially all the possible. Why values it can take on. So we see that Ah, we have all of the negative why values is an option. But the greatest value that why can take on is positive for and then after that, the graft turns around, starts going back down. So our range is actually going to be all. Why values less than or equal to positive? For now, we need to determine if the function is onto and on to function is a function such that for every, uh, possible value in the output set. There is some input that produces that output, and we could see that this function is not onto because there is no way to produce any of the output values greater than four from any of the input values. Like if I wanted to output five with this function, there would be no input that I could give because we&#x27;ve already established that the maximum value can take on its four. So now I need to determine if the function is 1 to 1 and one toe. One means that for every possible output, there&#x27;s only one input that gives that output so we can see that this function is also not one toe one. Because if I want to produce the output of zero, I can clearly either choose negative one as my input or positive three as my input. So the function is neither one toe, one nor

Finian Lickona · Answer

Okay, So for this exercise were given this graph and were asked to determine the roots or zeroes of thief function of what the range of the function is and whether or not is onto and or 1 to 1. So I&#x27;m gonna go ahead and write out the equation for this function. This is the graph of the function y equals x cubed. And I think it&#x27;s very easy to tell that the only X value that makes us equal to zero is going to be X equals zero. So the only real route that this function has is zero, but it&#x27;s a rule with multiplicity Three anyways. Ah, so now we need to determine the range of the function, which is all of the possible output values. And we see this extends from, uh, zero up to positive infinity and down a negative infinity. So our range is all really numbers. Now we need to determine if the function is onto and then on to function is a function such that for every output in the output set, there is some input that gives us that output. So if we wanted to output any of any, why value um there would be some X value we could choose to generate that output. So this is definitely an on to function. And now we want to see if this is a 1 to 1 function. So 121 means that for every output there is on Lee one input that produces that output. Ah, And if we look at this graph, we see that, for example, if we wanted to generate positive too, the Onley input that we could give would be something like, uh, it&#x27;s like one and some decimal value. It doesn&#x27;t matter. Anyways, the point is that, yes, this is a wonder one function because there&#x27;s only one way to generate a one choice of input to generate each possible output.

Finian Lickona · Answer

Okay, So for this exercise, we are given not quite. This graph the graph forgiven isn&#x27;t actually a polynomial that you could mathematically generate. So what I had to do was sort of approximated and then modify the function a little bit. And I had to do this so that the range answer would match up with the graph given in the book. But this is still effective enough for our purposes of answering the question. So the first thing we want to do is find all of the roots of this function, and that is pretty straightforward. Pretty easy to see that the roots are negative. Three negative one positive one in positive three. So we&#x27;re gonna write that in, and now we want to determine the range of the function, which is all of the possible output values it has. So, um, we can see from this graph that are why value can go down to negative one, but no further. This is the lowest point for both of these graphs. Yeah. Yes, that, um So ah, we see that the range is while all why values greater than or equal to negative one. Okay, so now we need to determine if this function is onto now. On on to function is any function where for every possible output, there&#x27;s some corresponding input, and we can see that there is no way to generate an output greater than negative one. So, for example, there is no input that is going to give us negative five as an output. So the function is not on two. Now we need to determine if the function is 1 to 1. Ah, one toe. One function is a function for which every possible output there is only one input that generates it. This is clearly not a wonder one function, because if I wanted to generate the output of five, I could either plug in. This looks like it&#x27;s a little bit more than three and a little bit less than negative three. And then over here I could pick some. Value is just a little bit more than three, and both of them generate the output of five. So this function is not ah, 1 to 1

Lily An · Answer

So for this one, it says, it tells us that the zeroes are at night of three and one. Well, we know for all polynomial sze zeros. If we want express that we can see x minus z one x minus E too Okay dot a dot depending on how many zeros are. So we know that if we have the zeros at negative three and one, if you put that in the X minus sect of three x minus one. So this becomes an X plus three. So it&#x27;s X plus three X minus one. And so, if we&#x27;re looking for the zeros, we just have to find when it touches at one and night of three where here&#x27;s one and here&#x27;s negative three So that one works. This one touches that one and negative three. So that one also works for that specific qualification. This one touches that negative one and positive three. So B does not work, and this one touches a three different points s O that also just not work. But then, if we look at our second qualification, it says that the range is less than or equal to four of the range His lesson or go to four. That means as before and down to here&#x27;s four we have points for and downs that one works, whereas see, it has negative for and up. So that one does not work. So is our answer.

Problem

In $3-11$ , each graph shows a polynomial functio…

Problem 3 Easy Difficulty

In $3-11$ , each graph shows a polynomial functions from the set of real numbers to the set of real numbers.
a. Find the real roots or zeros of the function from the graph, if they exist.
b. What is the range of the function?
c. Is the function onto?
d. Is the function one-to-one?
(GRAPH NOT COPY)

Answer

a) None
b) $\{3\}$
c) Onto
d) Not one-to-one

Related Courses

Algebra 2 and Trigonometry

Related Topics

Discussion

Top Algebra Educators

Catherine R.

Anna Marie V.

Alayna H.

Kristen K.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

Ann Xavier Gantert

Algebra 2 and Trigonometry

Catherine R.

Anna Marie V.

Alayna H.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

a) Noneb) $\{3\}$c) Ontod) Not one-to-one

Algebra 2 and Trigonometry

Catherine R.

Anna Marie V.

Alayna H.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

Ann Xavier GantertAlgebra 2 and Trigonometry

Catherine R.

Anna Marie V.

Alayna H.

Kristen K.

a) None
b) $\{3\}$
c) Onto
d) Not one-to-one

Ann Xavier Gantert

Algebra 2 and Trigonometry