Question

1 Using your function, explain to the Martians how to solve for f(3).

2. Describe to the Martians how to find the inverse of your function.

3. The Martians ask you to explain one last thing, Ultimate Math Ambassador. They ask you to create a new function, h(x). Then assign any number to x. Using complete sentences, explain whether f(h(x)) and h(f(x)) will always result in the same number. You will use the function f(x) that you created in problem number 2

My function is 4x+5=-3x+19

Answer 1

If your function is 4x+5=-3x+19, the logical next step would be to solve it for x (not for f(3)).

Add 3x to both sides and simplify the result.
Then subtract 5 from both sides and simplify the result. What do you get?

Answer 2

i did all that x=2
@mathmale

Answer 3

Happy to hear that. Please share your work on the remaining problems, so I can see what you've already done and what you still need to do. Here you end up with 7x=14, so, yes, x=2. That leaves it something of a mystery what is meant by "solve for f(3)."

Answer 4

What are your own thoughts on that?

Answer 5

i need help solving for f(3)

Answer 6

I understand that's what the question asks you to do. However, "your function" has neither f(x) nor y in i, so how are you going to evaluate f(3)?

Answer 7

@johnlegend : OpenStudy tells me you're looking at someone else's math problem instead of at your own. When you're done, let me know if you want to finish your math problem.

Answer 8

@mathmale i accidentally clicked it and was wondering why i was on it sorry @mathmale i didnt know i needed a y,because another person told me i didnt

Answer 9

You are given the function / relationship / equation 4x+5=3x+19. This can be solved for x, as you have already done, but does not involve f(x) or y.

However, we could CREATE OUR OWN FUNCTION, as follows:

Rewrite 4x+5=3x+19 by moving the two terms on the right side to the left side AND THEN equating the result to f(x). Please do this. Type out what you now have.

Answer 10

(Awaiting your response.)

Answer 11

so would it be 4+5x=3+19x @mathmale

Answer 12

I was asking you to start with that equation, but then to move the right two terms (the terms on the right side of your equation) to the left, and then labeling the result f(x). Please do that.


4 +5x = 3 + 19x
-3-19x = -3 -19x
------- --------

combine.
Set the resulting expression = to f(x).

Answer 13

@mathmale are we solving for f(3) still?

Answer 14

We will, but would you please finish this task first?

Answer 15

oh im sorry

Answer 16

no and im watching @mathmale

Answer 17

Cool. Remember, we found that x=2 is a solution to your original equation.
If we were to evaluate f(2), we'd get 7(2)-14=??

Try evaluating f(3).

Answer 18

f(x)=7x-14.

f(2)=7(2)-14 = ?

Answer 19

Please go ahead. If your screen says "mathmale is typing a reply," ignore it and type your own reply.

Answer 20

@mathmale 7 (3)-19 or is 7(3)-14 @mathmale

Answer 21

Hi, John, what's up at your end?

Answer 22

oh im sorry @mathmale you can continue

Answer 23

John:

f(x)=7x-14.
If we evaluate this at x=2, we get f(2)=7(2)-14=14-14=0.

If we evaluate this at x=3, we get what ??

Answer 24

John, are we having technical problems? I'd like for this discussion to move faster, if at all possible.

Answer 25

John: In the upper, left hand corner of your OpenStudy screen, you should see a symbol that looks like an envelope. This is for sending private messages. Send me a private message when you are ready to resume work on these problems that you've posted. In the meantime I'll be helping others.

Answer 26

@mathmale openstudy seems to be moving slow on my computer so if you dont see me replying thats why but f(3)-14=21-14=7

Answer 27

that's it. That's the answer you wanted.
Let's review this.

Answer 28

You were given an equation that had no f(x) or y in it.
You were asked to evaluate f(3).
This means that you had to create your own function from the given function.
Rearranging the given relationship by moving the 2 terms on the right over to the left side, and setting the result equal to f(x), we got f(x)=7x-14.

Answer 29

Note how f(2)=0. What this says, in effect, is, "Yes, 2 is a root of the original equation."

Now you can evaluate f(x) for any x.

Thus, f(3) = ?

Answer 30

f(3)=7

Answer 31

Right. And your function is f(x)=7x-14. How would you do Part 2 and find the inverse of this function? Have you done this type of problem before?

Answer 32

Finding the inverse of an integer means performing an operation that will cancel out the number

Answer 33

Let's paraphrase that: "Finding the inverse of a given FUNCTION results in obtaining a new FUNCTION that will cancel out the work of the original function."

Answer 34

If f(x)=7x-14,
replace f(x) with the label "y=" below:

Answer 35

ok @mathmale what is y?

Answer 36

I was just asking you to replace f(x) with y:
f(x)=7x-14 => y=7x-14.

Answer 37

Next, interchange x and y. Where you see x, write y; where you see y, write x. Result?

Answer 38

ok x=2

Answer 39

Take your y=7x-14. Replace that y with x and that x with y:

x=7y-14. This is what I'd asked you to do.

Answer 40

Now solve x=7y-14 for y. (This is a problem in algebra.)

Answer 41

Step 1: Add 14 to both sides of the equation and simplify the result. What do you get?
Type in your result.

Answer 42

x=21y @mathmale

Answer 43

Made a mistake; I'm sorry.

Here's what I'd do:
x = 7y - 14
+14= +14
--- ----------


Combine and simplify. Note: 7x + 14 does not equal 21 y. 7x and 14 are "unlike" terms and thus can't be combined.

Answer 44

i didnt know you meant to all sides im sorry @mathmale so is it 14x=7y-0

Answer 45

On the left side, x plus 14 equals what? Write this symbolically.

Answer 46

14x @mathmale

Answer 47

that looks like multiplication, not addition ("plus").
"x plus 14" could be written "14 added to x" and is represented symbolically by x+14.

thus, on the left side of your equation you have just x+14; on the right side, you have 7y -14 + 14, or 7y + 0, or just 7y.

Answer 48

In summary, we have x+14=7y, or

7y=x+14.

How would you solve that for y?

Answer 49

subtract 14 from both sides

Answer 50

7y indicates multiplication. The operation opposite to multiplication is division. Therefore, to isolate y on the left side (to solve for y) requires that we DIVIDE each side by 7:

\[7y=x+14\rightarrow \frac{ 7y }{ 7 }=\frac{ x }{ 7 }+\frac{ 14 }{ 7 }\]

Answer 51

Please simplify this as best you can. Note again: We are trying to solve for y.

Answer 52

Hint:\[\frac{ 7y }{ 7 }=y\]

Answer 53

therefore, our equation becomes\[y=\frac{ x }{ 7 }+\frac{ 14 }{ 2 }\]If you were running out of time, you could leave this equation as is; it would represent the INVERSE FUNCTION of the given function. But go ahead and reduce 14/7. What's that?

Answer 54

2

Answer 55

Yes, so that leaves us with the equation of the inverse function, in simplest form:\[y=f ^{-1}(x)=\frac{ x }{ 7 }+2\]

Answer 56

this completes the solution of Part 2. Questions? It's extremely important that you know how to do these operations yourself, so I'd strongly suggest that you review what we've done and then try to find the inverse of y=7x-14 yourself.

Answer 57

@johnlegend : If you wish to continue, let me know what you want to do next.

Answer 58

is x in the parentheses 2 @mathmale

Answer 59

Not sure I understand your question.

Your function, f (x), is 7x-14.

Your inverse function, the inverse of f(x)=7x-14, is
\[f ^{-1}(x)=\frac{ x }{ 7 }+2\] What are you asking?

Answer 60

do we have to solve the inverse function @mathmale

Answer 61

for Part 2, our job was to find the inverse function of f(x)=7x-14. We did. See the formula for the inverse function, above and below:\[f ^{-1}(x)=\frac{ x }{ 7 }+2\]

So, there's nothing to "solve."

What next?

Answer 62

part 3 @mathmale