Question

Find the inverse of
`c(x)=2log_5(x-1)` and find the domain of the inverse. Use interval notation to represent domain.

Answer 1

`c^(-1)(x)=5^{(x)/(2)}+1, \quad \textsf{for}\; \{x : x \in \mathbb{R} \}`

Explanation:

Given function:

`c(x)=2 \log_5(x-1)`

As the logarithm of zero or a negative number cannot be evaluated, the domain of the given function is restricted: (1, +∞).

The range of the given function is unrestricted: (-∞, +∞).

The inverse of a function is its reflection in the line y = x.

To find the inverse of a function, replace x with y:

`\implies x=2 \log_5(y-1)`

Rearrange the equation to make y the subject.

Divide both sides by 2:

`\implies\log_5(y-1)=(x)/(2)`

`\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b`

`\implies 5^{(x)/(2)}=y-1`

Add 1 to both sides:

`\implies y=5^{(x)/(2)}+1`

Replace y with c⁻¹(x):

`\implies c^(-1)(x)=5^{(x)/(2)}+1`

The domain of the inverse of a function is the same as the range of the original function.

Therefore, the domain of the inverse function is (-∞, +∞).

Therefore, the inverse of the given function is:

`c^(-1)(x)=5^{(x)/(2)}+1, \quad \textsf{for}\; \{x : x \in \mathbb{R} \}`