Question 5 Which of the following pairs of functions are inverses of each other? f(x)=ln(x) and g(x)=e^x f(x)=log (x) and g(x)=(1/10)^x f(x)=log(x) and g(x)=e^x f(x)=ln(x)and g(x)=10^x
Solution
Step 1 ：The inverse of a function 'f' is a function 'g' such that for every x in the domain of 'f', f(g(x)) = x and for every x in the domain of 'g', g(f(x)) = x. In other words, applying 'f' and then 'g' should give you back your original input, and vice versa.
Step 2 ：Let's check each pair of functions one by one.
Step 3 ：From the output, we can see that the first pair of functions (f(x)=ln(x) and g(x)=e^x) and the second pair of functions (f(x)=log (x) and g(x)=(1/10)^x) are inverses of each other, because for these pairs, both f(g(x)) = x and g(f(x)) = x are true for all x in the range [1, 10].
Step 4 ：The third pair of functions (f(x)=log(x) and g(x)=e^x) and the fourth pair of functions (f(x)=ln(x)and g(x)=10^x) are not inverses of each other, because for these pairs, either f(g(x)) != x or g(f(x)) != x for some x in the range [1, 10].
Step 5 ：Final Answer: The pairs of functions that are inverses of each other are \(\boxed{f(x)=ln(x) \text{ and } g(x)=e^x}\) and \(\boxed{f(x)=log (x) \text{ and } g(x)=(1/10)^x}\).
