Composition of Functions

In summary, the conversation discusses the function f(x)= x/(1+x) and finding the domain of f(f(x)). The attempt at a solution found f(f(x))= x/(1+2x) and initially, the domain was determined to be (-∞,-1/2)∪(-1/2,∞). However, it was pointed out that f(f(-1)) is undefined, so the correct domain is (-∞,-1)∪(-1,-1/2)∪(-1/2,∞). The process for finding the domain was also discussed using the definition D: {x∈(-∞,-1)∪(-1,∞) | (
  • #1
84
2

Homework Statement


f(x)= x/(1+x)

What is f(f(x)) and what is its domain.

2. The attempt at a solution
I found f(f(x))= x/(1+2x)
and the domain: (-∞,-1/2)∪(-1/2,∞) , but it is saying that I have the wrong domain. What mistake have I made?


My process for finding domain:
1. Find the domain of f : x≠-1
2. Use the definition D: {x∈(-∞,-1)∪(-1,∞) | (x/1+x) ≠ -1}
3. Find the ∩ (intersection) for the two domains.
 
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  • #2
Jpyhsics said:

Homework Statement


f(x)= x/(1+x)

What is f(f(x)) and what is its domain.

2. The attempt at a solution
I found f(f(x))= x/(1+2x)
and the domain: (-∞,-1/2)∪(-1/2,∞) , but it is saying that I have the wrong domain. What mistake have I made?


My process for finding domain:
1. Find the domain of f : x≠-1
2. Use the definition D: {x∈(-∞,-1)∪(-1,∞) | (x/1+x) ≠ -1}
3. Find the ∩ (intersection) for the two domains.

What is ##f(f(-1)##?
 
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  • #3
PeroK said:
What is ##f(f(-1)##?
Oh I see! Its undefined!
So I guess my domain should be (-∞,-1)∪(-1,-1/2)∪(-1/2,∞)

Thank You!
 
  • Like
Likes Delta2
  • #4
Jpyhsics said:
Oh I see! Its undefined!
So I guess my domain should be (-∞,-1)∪(-1,-1/2)∪(-1/2,∞)

Thank You!

Formally: if ##D_1 = A \cup B## and ##D_2 = C \cup D## then $$ D_1 \cap D_2 = (A\cap C) \cup (A \cap D) \cup (B \cap A) \cup (B \cap D)$$ Apply this to ##A = (-\infty,-1),## ##B = (-1,\infty)##, ##C = (-\infty, -1/2)## and ##D = (-1/2,\infty).##
 

What is the definition of composition of functions?

The composition of functions is a mathematical operation where two functions are combined to create a new function. It is denoted as (f ∘ g)(x) and read as "f composed with g of x". The output of one function is used as the input for the other function.

What is the domain and range of a composite function?

The domain of a composite function is the set of all values that can be input into the initial function and produce a valid output that can be used as an input for the second function. The range of a composite function is the set of all values that can be output from the final function.

How is composition of functions evaluated?

The composition of functions is evaluated by plugging the inner function into the outer function. For example, if f(x) = 3x and g(x) = x + 2, then (f ∘ g)(x) = f(g(x)) = 3(x + 2) = 3x + 6. The inner function (g(x)) is substituted into the outer function (f(x)), and then the resulting expression is simplified.

What is the difference between composition and multiplication of functions?

Composition of functions combines two functions to create a new function, whereas multiplication of functions multiplies two functions to create a new function. In composition, the output of one function becomes the input of the other, while in multiplication, the two functions are multiplied together.

What are some real-world applications of composition of functions?

Composition of functions is used in various fields such as engineering, economics, and computer science. It can be used to model complex systems, optimize processes, and analyze data. For example, in economics, the demand curve for a product can be represented as a composition of two functions - one for the price and one for the quantity demanded.

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