Question

How do you find the inverse of \[y = \log \left( {x + 4} \right)\]?

Answer 1

Hint:
An inverse function is defined as a function, which can reverse into another function. If f and g are inverse functions, then \[f\left( x \right) = y\] if and only if \[g\left( y \right) = x\] and to find an inverse function, just interchange the x and y terms of the given function.

Complete step by step solution:
Let us write the given function
\[y = \log \left( {x + 4} \right)\]
To find the inverse, just interchange the x and y terms as
\[x = \log \left( {y + 4} \right)\]
We need to solve for y
\[x = {\log _{10}}\left( {y + 4} \right)\]
As we know that \[\log \left( {y + 4} \right)\]can also be written as \[{\log _{10}}\left( {y + 4} \right)\].
\[{10^x} = {10^{\log 10\left( {y + 4} \right)}}\]
Simplifying the log function, we get
\[{10^x} = y + 4\]
As we need to find the inverse of y, hence we get
\[y = {10^x} - 4\]
Therefore, the inverse of \[y = \log \left( {x + 4} \right)\] is
\[y = {10^x} - 4\].

The value of x is accompanied by only one value of y, so this relation is a function.

Additional information:
The properties of inverse functions are listed and discussed below.
Only one to one function has inverses i.e., If g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other.
If f and g are inverses of each other than both are one to one function.

Note:
A function normally implies what y is if you know what x is. The inverse of a function will imply what x had to be to get that value of y. For a function that is defined to be the set of all ordered pairs (x, y), the inverse of the function is the set of all ordered pairs (y, x). The domain of the function becomes the range of the inverse of the function. The range of the function becomes the domain of the inverse of the function.