## anonymous one year ago Given b(x)=|x+4|, what is b(-10)? A.-10 B. -6 C. 6 D. 14

1. SolomonZelman

at first, plug in -10 for x, into the $$|x+4|$$.

2. anonymous

-6?

3. SolomonZelman

no, hold on, when you plug in -10 for x, you get: $$\small b(x)=\left|x+4\right|$$ $$\small b(\color{red}{-10})=\left|\color{red}{-10}+4\right|$$ $$\small b(\color{red}{-10})=\left|-6\right|$$

4. SolomonZelman

What does "absolute value of a", written as "|a|" mean? Roughly speaking, you can define |a| as a distance from 0 to a (how much do you have to walk from a to 0)

5. anonymous

so would it be 6 instead of negative 6?

6. SolomonZelman

|dw:1437934269242:dw|

7. SolomonZelman

So, yes the answer is 6.

8. SolomonZelman

in other words, the distance from 0 to -6 is 6. why is it so that 0-6=-6 (not just 6) because the minus by 6 indicates the direction (that it is going to the left - if you look at the number line), but absolute value of -6, or |-6| is 6

9. SolomonZelman

So, lets review the work we have to do:

10. SolomonZelman

$$\small b(x)=\left|x+4\right|$$ $$\small b(\color{red}{-10})=\left|\color{red}{-10}+4\right|$$ $$\small b(-10)=\left|-6\right|$$ $$\small b(-10)=-6$$

11. SolomonZelman

if you want to ask anything you are always welcome to do so...:)

12. anonymous

That helps a lot! Thank you so much! Would you be able to help with another problem?

13. anonymous

@SolomonZelman

14. SolomonZelman

Yes, perhaps, but I am helping another student currently. I personally don't care tho', if you post it in this thread or another one, so you can keep this post.

15. anonymous

For which pair of functions is (f ° g)(x) =x? A) f(x) =x^2 and g(x) = 1/x B) f(x)=2/x and g(x) = 2/x C) f(x)= x-2/3 and g(x)= 2-3x D) f(x)=1/2x-2 and g(x)= 1/2x+2

16. SolomonZelman

I will show you an example on two different functions.

17. SolomonZelman

So, lets say my function f(x) and g(x) are the following: $$\large\color{blue}{ \displaystyle f(x)=3x^3+2 \\[0.5em]}$$ $$\large\color{red}{ \displaystyle g(x)=\frac{5}{x} }$$ -------------------------------------------- A notation of: $$(\color{blue}{f}^\circ \color{red}{g})(x)$$ means the same as $$\color{blue}{f(~\color{red}{g(x)}~)}$$ in other words, you take a function g(x), and plug it instead of x, into the f(x). -------------------------------------------- How would it actually go overhere? $$\large\color{black}{ \displaystyle (\color{blue}{f}^\circ \color{red}{g})(x)=\color{blue}{f(~\color{red}{g(x)}~)}=\color{blue}{3\left(\color{red}{\frac{5}{x}}\right)^3+2} }$$ see how I am plugging the g(x) instead of x, into f(x)? now, we will simplify that: $$\large\color{black}{ \displaystyle (\color{blue}{f}^\circ \color{red}{g})(x)=\color{blue}{f(~\color{red}{g(x)}~)}=3\left(\frac{5}{x}\right)^3+2 }$$ $$\large\color{black}{ \displaystyle (\color{blue}{f}^\circ \color{red}{g})(x)=\color{blue}{f(~\color{red}{g(x)}~)}=3\frac{5^3}{x^3}+2 }$$ $$\large\color{black}{ \displaystyle (\color{blue}{f}^\circ \color{red}{g})(x)=\color{blue}{f(~\color{red}{g(x)}~)}=3\frac{125}{x^3}+2 }$$ $$\large\color{black}{ \displaystyle (\color{blue}{f}^\circ \color{red}{g})(x)=\color{blue}{f(~\color{red}{g(x)}~)}=\frac{375}{x^3}+2 }$$

18. SolomonZelman

so all you are doing for (fº g)(x), is that you plug in the function g(x) instead of x --> into the f(x).

19. SolomonZelman

my function is more rather complicated, but all you need to do is to find this (fº g)(x), and which ever of the options will give you a result/answer of x, that is your answer.

20. SolomonZelman

I will respot the question: ---------------------- For which pair of functions is (f ° g)(x) =x? A) f(x) =x^2 and g(x) = 1/x B) f(x)=2/x and g(x) = 2/x C) f(x)= x-2/3 and g(x)= 2-3x D) f(x)=1/2x-2 and g(x)= 1/2x+2

21. SolomonZelman

now, lets find (fº g)(x) for each answer choice together: which one do you want to start from?

22. anonymous

Which do you think would work better?

23. anonymous

We can just start from A

24. SolomonZelman

yes, lets do that.

25. SolomonZelman

f(x) =x² g(x) = 1/x (fº g)(x) = f(g(x)) = (1/x)²=1²/x²=1/x²

26. SolomonZelman

so is A the answer or not?

27. anonymous

no!

28. SolomonZelman

yes, it is not the answer.

29. SolomonZelman

lets do the next one, but this one i will ask you to take a shot on

30. SolomonZelman

f(x)=2/x g(x) = 2/x (fº g)(x) = ? (go ahead)

31. anonymous

(2/x)^2 =x^2=2/x^2 is that how we set it up

32. SolomonZelman

(fº g)(x) is same as f(g(x)) so you just plug in the g(x) (in this case, 2/x), instead of x, into the f(x). so f(x) = 2/x (fº g)(x) = f(g(x))=2/(2/x)

33. SolomonZelman

all we did is that we took the x in f(x), and replaced it by the function g(x). (i.e. replaced the x in the function f(x), but 2/x)

34. SolomonZelman

now, you need to simplify this. can you do that?

35. anonymous

I got 1/x is that correct?

36. SolomonZelman

|dw:1437937373887:dw|

37. SolomonZelman

|dw:1437937442588:dw|

38. anonymous

then would it just be x? plain and simple?

39. SolomonZelman

yes

40. SolomonZelman

it is =x

41. SolomonZelman

and thus B is the answer

42. anonymous

AHA! Thank you so much