
a.
If f(x)=6x then find f−1(x) .
a.

Answer to Problem 33WE
The inverse of function f(x)=6x is f−1(x)=log6x .
Explanation of Solution
Given information: The given function is,
f(x)=6x....(1)
Calculation: For the given function f(x)=6x , the inverse is evaluated as,
y=6x....(Plugging f(x)=y)or, x=6y....(Exchanging variables)y=log6x....(Solving for variable (y))f−1(x)=log6x....(Plugging y=f−1(x))....(2)
Hence, the inverse of function f(x)=6x is f−1(x)=log6x .
b.
Find the value of f−1(36) and f−1(1√6) for the function f(x)=6x
b.

Answer to Problem 33WE
The value of f−1(36) is 2 and f−1(1√6) is −12 .
Explanation of Solution
Given information: The given function is f(x)=6x .
Calculation: The value of f−1(36) and f−1(1√6) can be evaluated by substitute x=36 and x=1√6 in (2). Therefore from (2),
f−1(36)=log6(36)....(For x=36)=log6(6)2 =2log66....(∵logaa=1)=2(1)....(∵logaa=1)=2
Similarly evaluate f−1(1√6) ,
f−1(1√6)=log6(1√6)....(For x=1√6)=log6(6)−12 =−12log66=−12(1)....(∵logaa=1)=−12
Hence the value of f−1(36) is 2 and f−1(1√6) is −12 .
c.
Find the domain and range of f(x)=6x and f−1(x) .
c.

Answer to Problem 33WE
The domain of function is {x|x∈(−∞,∞)} and range of function is {y|y∈(0,∞)} whereas the domain of f−1(x)=log6x is {x|x∈(0,∞)} and range of is {y|y∈(−∞,∞)} .
Explanation of Solution
Given information: The given function is f(x)=6x .
Calculation: Since the given function is f(x)=6x and from (2) f−1(x)=log6x . Therefore domain (Df) of function f(x)=6x is {x|x∈(−∞,∞)} and range (Rf) of f(x)=6x is {y|y∈(0,∞)} . And domain (Df−1) of f−1(x)=log6x is {x|x∈(0,∞)} and range (Rf−1) of f−1(x)=log6x is {y|y∈(−∞,∞)} .
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Algebra and Trigonometry: Structure and Method, Book 2
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