Answer:
[tex]c^{-1}(x)=5^{\frac{x}{2}}+1, \quad \textsf{for}\; \{x : x \in \mathbb{R} \}[/tex]
Step-by-step explanation:
Given function:
[tex]c(x)=2 \log_5(x-1)[/tex]
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:
[tex]\implies x=2 \log_5(y-1)[/tex]
Rearrange the equation to make y the subject.
Divide both sides by 2:
[tex]\implies\log_5(y-1)=\dfrac{x}{2}[/tex]
[tex]\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b[/tex]
[tex]\implies 5^{\frac{x}{2}}=y-1[/tex]
Add 1 to both sides:
[tex]\implies y=5^{\frac{x}{2}}+1[/tex]
Replace y with c⁻¹(x):
[tex]\implies c^{-1}(x)=5^{\frac{x}{2}}+1[/tex]
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:
[tex]c^{-1}(x)=5^{\frac{x}{2}}+1, \quad \textsf{for}\; \{x : x \in \mathbb{R} \}[/tex]
