Compositions and inverse of of functions

In summary, a composition of functions is when one function's output becomes the input for another function. To find the composition of two functions, you simply substitute the inner function's output into the outer function. The inverse of a function is a function that "undoes" the original function by switching the input and output values. To find the inverse, switch the x and y variables and solve for y. The relationship between compositions and inverses of functions is that the composition of a function and its inverse will always result in the original input value. This is because the inverse "undoes" the original function, cancelling it out when the two are composed.
  • #1
drop
14
0
Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :)

Part B

2) Given the functions f(X) = 7x^2 - 5x and g(x) = 2x - 3 determine and simplify the following:

a) (f-g)(x)

My Answer: 7x^2 - 7x - 3

b) (f - g)(2)

My Answer: 17

c) (fg)(x)

My Answer: 14z^3 - 31x^2 + 5x

d) g^-1(x)

My Answer: 1/2x + 3/2
 
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  • #2
Re: Please check my answers - 8

Part a) is incorrect, but it must merely be a typo since you have the correct result for part b). The sign of the third term is wrong.

Part c) is incorrect...you have a typo in which $z$ is where $x$ should be in the first term, and the last term is incorrect, but I suspect this is also a typo.

Part d) is correct.

I don't want to discourage you from posting, and we really do want to help you when you get stuck or need guidance, insights, etc. but for simply checking your results, it may be quicker for you to use a site such as:

Wolfram|Alpha: Computational Knowledge Engine

Many of the problems you posted can be entered there and the answer gotten immediately, to see if your end result is correct. This site can plot functions, find extrema, inverses, simplify expressions, etc.
 
  • #3
Re: Please check my answers - 8

Can you tell me what I did wrong in 3a?

I took f(x) minus g(x) and wrote it like this:
7x^2 - 5x + 0
0x^2 + 2x - 3
7x^2 - 7x - 3

and thank you for that link.
 
  • #4
Re: Please check my answers - 8

drop said:
Can you tell me what I did wrong in 3a?

I took f(x) minus g(x) and wrote it like this:
7x^2 - 5x + 0
0x^2 + 2x - 3
7x^2 - 7x - 3

and thank you for that link.

I would write:

\(\displaystyle (f-g)(x)=\left(7x^2-5x \right)-\left(2x-3 \right)=7x^2-5x-2x+3=7x^2-7x+3\)

You see, when you subtract, you essentially change all the signs, then add. Subtracting a negative is the same as adding a positive.

Or to use your notation:

7x^2 - 5x + 0
0x^2 + 2x - 3
7x^2 - 7x + 3
 
  • #5
Re: Please check my answers - 8

OH yeah, haha I knew that :P Thanks!
 
  • #6
Re: Please check my answers - 8

MarkFL said:
Part c) is incorrect...you have a typo in which $z$ is where $x$ should be in the first term, and the last term is incorrect, but I suspect this is also a typo.

Yep, that was a big typo, sorry. Actually what I had written on my paper was:

14x^3 - 31x^2 + 15x

Would that answer be correct?
 
  • #7
Re: Please check my answers - 8

drop said:
Yep, that was a big typo, sorry. Actually what I had written on my paper was:

14x^3 - 31x^2 + 15x

Would that answer be correct?

Yes, that's correct.
 

Related to Compositions and inverse of of functions

1. What is a composition of functions?

A composition of functions is when one function's output becomes the input for another function. It is denoted by (f o g)(x), which means f(g(x)).

2. How do you find the composition of two functions?

To find the composition of two functions, simply substitute the inner function's output into the outer function. For example, if f(x) = 2x and g(x) = x+3, then (f o g)(x) = f(g(x)) = 2(x+3) = 2x + 6.

3. What is the inverse of a function?

The inverse of a function is a function that "undoes" the original function. It switches the input and output values, so if f(x) = y, then f-1(y) = x.

4. How do you find the inverse of a function?

To find the inverse of a function, switch the x and y variables and solve for y. If the original function is one-to-one (each input has a unique output), then the inverse will also be a function. If the original function is not one-to-one, then the inverse will be a relation.

5. What is the relationship between compositions and inverses of functions?

The composition of a function and its inverse will always result in the original input value. In other words, (f o f-1)(x) = x and (f-1 o f)(x) = x. This is because the inverse "undoes" the original function, so when the two are composed, they cancel each other out.

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