Composition of Functions: Given f(x)= 1/(x-2) and g(x)= (3/x)+2; find f(g(x))

Answer 1

To find f[g(x)] plug the g function into the f function:

f[g(x)] = 1/[(3/x +2) - 2]= 1/(3/x) = x/3.

Take a look here to confirm:

desmos.com/calculator/ht1csap8q1

Answer 2

f(x) = 1/(x-2) g(x) = (3/x)+2

f(g(x)) = 1/((3/x)+2)-2) = 1/(3/x) = x/3

Answer 3

Answer 4

For this problem, its easier if you understand what functional notation means in english:

f(3) means the same as "function at x=3"

So going by this, if you wanted to solve f(3), you would just take 3 and substitute it for x in the formula:

f(x) = 1/(x-2). This translates to 1/(3-2) which is just 1/1 or simply 1.

Do you see the pattern here? You substitute what is given to you inside f(x) into wherever it says x inside the formula,

So, now let us think what you are being asked to calculate.

You are asked to calculate f(g(x))

So what is inside? Well, inside f is g(x). So you just substitute g(x) into the formula like this:

1/(x-2) becomes

1/(g(x) - 2)

Now we know what g(x) is, so we can substitute that in as well:

1/((3/x) +2 - 2)

this becomes:

1/(3/x) which is the same as x/3.

So f(g(x)) = x/3

Answer 5

Answer 6

Hi! For this problem, it looks like you're being assessed on your ability to combine different functions and create a new, "combined" result. Luckily, I can help!

For equations such as f(g(x)), the function inside the parenthesis-- in this case: g(x) -- is the equation you're going to plug in in place of x. The variable outside-- f --is the original equation.

In this example, we are given 2 separate, defined functions:

f(x) = 1/(x-2) &

g(x)= (3/x)+2

Using this given information, we can use a simple process to combine their equations and find f(g(x)). Please note that we are largely going to disregard "f(x)" and "g(x)" as descriptions of our two starting functions, and instead, we will be referring to them in terms of x (I.e. "1/(x-2)" instead of "f(x)" and "(3/x)+2" instead of "g(x)").

Step 1: Take the function f(x) --> this is like your template.

In this problem, we're given that f(x)=1/x-2. Therefore, we will start with 1/x-2

Step 2: For the x value, swap in the g(x) function in its place (x -> (3/x)+2)

1. In step 1, we found that our "template" expression was 1/x-2. Now, we are going to replace the "x" by substituting in what we are given for g(x). Since g(x) = (3/x)+2, this is what we will be using for the substitution.
2. f(g(x))= 1/((3/x+2)-2   Every other part of f(x) stays the same. Note that instead of simply writing "x," it has been replaced with the full expression of g(x).

Step 3: Solve the integrated, or composite, equation.

1. Doing so should yield an answer of x/3. This is your final answer.

Answer: x/3

If you have any questions or would like additional guidance on other problems, please do not hesitate to reach out, and I would be happy to help!