Inverse Functions: Definition, Formulas and Example Problems
Inverse Functions: Definition, Formulas and Example Problems – What is meant by the inverse function? At this time About the knowledge.co.id will discuss about the inverse function and the things that surround it. Let's look at the discussion together in the article below to better understand it.
Inverse Functions: Definition, Formulas and Example Problems
The inverse function occurs because there is a function that is denoted by f (x) and has a relationship in each set A to each set B.
So it will be an inverse function denoted by f-1 (x) which has nothing but a relation from set B to every set A.
Thus, the inverse function is obtained from f: A → B which changes to f-1 B → A so that the origin of or the domain f (x), becomes the friendly region or the codomain becomes the result area or range f-1 (x), namely the set A. And vice versa for set B.
The inverse function or also known as the reverse function is a function that is the opposite of the original function.
A function f has an inverse function f
-1 if f is a one-to-one function and a function on (bijective). This relationship can be stated as follows:(f-1)-1 = f
Simply put, a bijective function takes place when the number of domain members is equal to the number of codomain members.
No two or more distinct domains map to the same codomain. And in each codomain has a partner in the domain. Pay attention to the pictures below:
Based on the image from the mapping above, the first mapping shows the bijective function.
The second mapping is not a bijective function because it only takes place on the on function.
Domains d and e map to members of the same codomain. The third mapping is not a bijective function because the mapping only takes place in one-to-one functions. Kodomain 9 has no partner in the member domain.
For example, f is a function that maps x to y, so we can write it as y = f (x), then f-1 is a function that maps y to x, written x = f-1(y).
For example f: A →B is a bijective function. The inverse function f is a function that combines each element of B with exactly one element of A.
The inverse function f is also represented by f-1 as follows:
There are 3 stages to determine the inverse function, among others:
- Change the form y = f (x) to form x = f (y).
- Write x as f-1(y) to f-1(y) = f(y).
- Replace the variable y with x so that you get the formula for the inverse function f-1(x).
In the inverse function there is a special formula like the following:
How to Find the Inverse of a Function
The way to find the inverse of a function is, for example, the function y=f (x) can be found in the following way:
- Make changes in the equation y=f (x) to the form x=f (y).
- Then replace x with f-1(y) so it becomes f(y) = f-1(y).
- If you change y with x, you will find the inverse of f (x) in the form f-1.
Inverse Functions in Life
In the following, we will provide examples of inverse functions that exist in everyday life, including:
- In the Field of Economics
The inverse function is used in calculating as well as estimating something, for example the function of demand and supply. - In the Field of Chemistry
The ivers function is used to determine the decay time of an element. - In the Field of Geography and Sociology
The inverse function is achieved in optimization in industry and also population density. - In Physics
The inverse function is used for quadratic function equations in explaining a motion phenomenon.
Examples of Inverse Function Problems
Problem 1
A mapping f: R→R with (g ◦ f)(x) = 2×2 + 4 x + 5 and g (x) = 2x + 3. Then f (x)=…
x2 + 2x + 1
x2 + 2x + 2
2x2 + x + 2
2x2 + 4x + 2
2x2 + 4x + 1
Answer:
Define f(x)
(g ◦ f)(x) = 2×2 + 4x + 5
g (f(x)) = 2×2 + 4x + 5
2(f (x)) + 3 = 2×2 + 4x + 5
f(x) = x2 + 2x + 1
The answer: A
- Question 2
If g (x – 2) = 2x – 3 and (f ◦ g)(x – 2) = 4×2 – 8x + 3, then f(-3) =…
-3
0
3
12
15
Answer:
g(x – 2) = 2x – 3
(f ◦ g)(x – 2) = 4×2 – 8x + 3
f (g(x – 2)) = 4×2 – 8x + 3
f (2x – 3) = 4×2 – 8x + 3
Define f(-3)
If -3 = 2x – 3 then x = 0
So that:
f(-3) = 4(0)2 – 8(0) + 3 = 3
The answer: A
Problem 3.
Let f: R→ R and g: R→R, f (x) = x + 2 and (g ◦ f)(x) = 2×2 + 4x – 6, Also suppose that x1and x2 are roots of g ( x) = 0 then x1 + 2×2 =…
0
1
3
4
5
Answer:
Define g(x).
(g ◦ f)(x) = 2×2 + 4x – 6
g (f(x)) = 2×2 + 4x – 6
g(x+2) = 2×2 + 4x -6
g (x) = 2(x – 2)2 + 4(x – 2) – 6 = 2×2 – 8x + 8 + 4x – 8 – 6 = 2×2 – 4x – 6
Define x1 + 2×2
g(x) = 0
2×2 – 4x – 6 = 0
x2 – 2x – 3 = 0
(x-3)(x+1) = 0
x1=3 →x2 = -1, so 3
x1 = 2×2 = 3+2 (-1) = 1
or
x1 = -1 → x2 = 3, so
x1 + 2×2 = (-1) + 2(3) = 5
The answer: E
- Problem 4
Find the inverse of the function F(x) = (2x + 2)2 – 5 ?
the usual way
Let F(x) = y
y = (2x + 2)2 – 5
y + 5 = (2x + 2)2
(y + 5)1/2 = 2x + 2
(y + 5)1/2 – 2 = 2x
[(y +5)1/2 – 2]/2 = x
Then f-1(x) = [(x + 5)1/2 – 2]/2
Alternative way
operation x on the function F(x) = (2x + 2)2 – 5 :
Multiply by 2
Plus 2
Squared
Minus 5
do it in reverse and in the following order:
Plus 5
A rank 2 root
Minus 2
divided by 2
The inverse result is f-1(x) = [(x + 5)1/2 – 2]/2
Thus the review from About the knowledge.co.id about Inverse Function, hopefully can add to your insight and knowledge. Thank you for visiting and don't forget to read other articles
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