Find the Domain f(4)f^-1(625)

Problem

ƒ(x)=5, find *ƒ(4)⋅ƒ(625)(−1)

Solution

1. Evaluate the function ƒ(x) at x=4 by substituting the value into the exponential expression.

ƒ(4)=5

ƒ(4)=625

2. Identify the inverse function ƒ(x)(−1) by recognizing that the inverse of an exponential function ax is the logarithm (log_a)(x)

ƒ(x)(−1)=(log_5)(x)

3. Evaluate the inverse function at x=625 by determining the power to which 5 must be raised to get 625.

ƒ(625)(−1)=(log_5)(625)

5=625⇒(log_5)(625)=4

4. Multiply the two results together to find the final value of the expression.

ƒ(4)⋅ƒ(625)(−1)=625⋅4

625⋅4=2500

Final Answer

ƒ(4)⋅ƒ(625)(−1)=2500

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