Question

Given the function f(x) = log base 4(x+8) , find the value of f^-1(2)

Answer 1

The value of
`f^(-1)(2)=8`

Explanation:

* Lets revise how to find the inverse function

- At first write the function as y = f(x)

- Then switch x and y

- Then solve for y

- The domain of f(x) will be the range of f^-1(x)

- The range of f(x) will be the domain of f^-1(x)

* Now lets solve the problem

- The inverse of the logarithmic function is an exponential function

∵
`f(x)=log_(4)(x + 8)`

- Write the function as y = f(x)

∴
`y=log_(4)(x+8)`

- Switch x and y

∴
`x=log_(4)(y+8)`

- Lets solve it to find y

# Remember:
`log_(a)b=n=====a^(n)=b`

- Use this rule to find y

∴
`4^(x)=(y + 8)`

- Subtract 8 from both sides

∴
`4^(x)-8=y`

∴
`f^(-1)(x)=4^(x)-8`

- Lets substitute x by 2

∴
`f^(-1)(2)=4^(2)-8`

- The value of 4² = 16

∴
`f^(-1)(x)=16-8=8`

* The value of
`f^(-1)(2)=8`