Question

Write the inverse of f(x) = 3x - 1

f -1(x) =

log 3 x + 1
log 3 ( x - 1)
log 3 x - 1

Answer 1

see below

Explanation:

we need to write the inverse of ,

`\longrightarrow f(x) = 3x -1 \\`

Substitute
`f(x)=y` ,

`\longrightarrow y = 3x-1 \\`

Interchange x and y ,

`\longrightarrow x = 3y -1 \\`

now solve for y ,

`\longrightarrow 3y = x + 1 \\`

`\longrightarrow y =(x+1)/(3) \\`

replace
`y` with
`f^(-1)(x)` as ,

`\longrightarrow \underline{\underline{ f^(-1)(x) =(x+1)/(3)}}\\`

And we are done!

Answer 2

`f^(-1)(x)=\log_3(x)+1`

Explanation:

Given function:

`f(x)=3^(x-1)`

To find the inverse of the given function, swap y and x:

`\implies x=3^(y-1)`

Take the log base 3 of each side of the equation:

`\implies \log_3(x)=\log_(3)3^(y-1)`

`\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax`

`\implies \log_3(x)=(y-1)\log_(3)3`

`\textsf{Apply log law}: \quad \log_aa=1`

`\implies \log_3(x)=(y-1)(1)`

`\implies \log_3(x)=y-1`

Add 1 to both sides of the equation:

`\implies \log_3(x)+1=y`

`\implies y=\log_3(x)+1`

Replace y for f⁻¹(x):

`\implies f^(-1)(x)=\log_3(x)+1`