Content:
Definitions
What is the Inverse of a function?What is the Domain of a function?
What is the Range of a function?
How to find the Inverse of a function
Example
Definitions:
What is the Inverse of a function?
On graph, f(x) and f -1(x) are symmetric about x = y line.
Explanation 1:
Suppose there is a function f(x): f(a) = b. And another function exist g(x): g(b) = a. Then g(x) can be called as inverse of f(x).
Explanation 2:
Let us consider a function of x: f(x). We can consider y = f(x). If somehow we can calculate x in terms of y then obtained function of y is f -1(y). Later by changing y with x we will get f -1(x).
Explanation 3:
If there is a function having sets of domain and range. For each element in the domain, there exist at least one element in the range. Let there exist a second function in which the domain is the same as the range of the first function and the range of this function is the same as the domain of the first function. If we put input of the first function a then we get some outputs, if we put these outputs as input of the second function then we get the same output as we put input for the first function.
What is the Domain of a function?
The domain of a function is a set of all possible values for which function can give at least one real number as output.
For example,
1) In 1/x, if x = 0 then it will give infinity as output so its domain does not contain 0.
2) In log(x), if x is negative then we don't get any output so its domain doesn't contain any negative values.
3) In arcsin(x) or arccos(x), if -1 ≤ x ≤ 1 then only we will get some output, so its domain will be [-1, 1].
What is the Range of a function?
The range of a function is a set of all possible values that can be obtained by function by giving all possible inputs.
For example:
1) sin(x) and cos(x) cannot give values less than -1 or values greater than 1, so its range only contain values from -1 to 1.
2) x2 only gives positive values, so its range will be 0 to infinity.
3) Exponential function will always be positive so its range will be [0, ∞).
How to find the inverse of a function?
To find the inverse of f(x),
1) Consider y = f(x).
2) Try to simplify for x. In other words, find the value of x in terms of y.
3) You will get x as a function of y. Let function of y is g(y).
4) Here g(y) = f -1(y).
5) Replace y with x to get f -1(x).
Example:
Solution:
Let, (Put y = f(x))
\[y=\frac{7^x-7^{-x}}{2}\]
Now solve for x
\[\Rightarrow y=\frac{7^{2x}-1}{2\cdot7^x}\]
\[\Rightarrow 7^{2x}-2\cdot7^x\cdot y -1 = 0\]
Its a quadratic equation,
\[\Rightarrow 7^x = \frac{2y \pm \sqrt{4y^2+4}}{2}\]
\[\Rightarrow 7^x = y \pm \sqrt{y^2+1}\]
Here, due to exponential function, the value should always positive;
\[\Rightarrow 7^x = y + \sqrt{y^2+1}\]
\[\Rightarrow x = \log_7 \left(y + \sqrt{y^2+1}\right)\]
Here, x is inverse function of f(y);
\[\Rightarrow f^{-1}(y) = \log_7 \left(y + \sqrt{y^2+1}\right)\]
Relace all y with x in above equation:
\[\Rightarrow f^{-1}(x) = \log_7 \left(x + \sqrt{x^2+1}\right)\]
