Final answer:
To determine if two functions are inverses, we need to check if the composition of the functions results in the identity function. Based on this, the pairs of functions that are inverses of each other are b) f(x) = x + 10 and g(x) = 4x - 10, and d) f(x) = 6x^2 - 7 and g(x) = x + 7.
Step-by-step explanation:
To determine whether two functions are inverses of each other, we need to check if the composition of the functions results in the identity function. In other words, if f(g(x)) = x and g(f(x)) = x for all x in the domain. Let's check each pair of functions:
a) f(x) = 2x + 9 and g(x) = 12(x-9).
First, let's find f(g(x)): f(g(x)) = 2(12(x-9)) + 9 = 24x - 18 + 9 = 24x - 9.
We can see that f(g(x)) is not equal to x. Therefore, a) is not a pair of inverse functions.
b) f(x) = x + 10 and g(x) = 4x - 10.
Similarly, let's find f(g(x)): f(g(x)) = (4x - 10) + 10 = 4x.
And g(f(x)): g(f(x)) = 4(x + 10) - 10 = 4x.
Both f(g(x)) and g(f(x)) are equal to x, so b) is a pair of inverse functions.
c) f(x) = 12 - 18 and g(x) = 18 - 12.
Once again, let's find f(g(x)): f(g(x)) = 12 - 18 = -6.
We can see that f(g(x)) is not equal to x. Therefore, c) is not a pair of inverse functions.
d) f(x) = 6x^2 - 7 and g(x) = x + 7.
For the last pair, let's find f(g(x)): f(g(x)) = 6(x + 7)^2 - 7.
And g(f(x)): g(f(x)) = f(g(x)) = (6x^2 - 7) + 7 = 6x^2.
Both f(g(x)) and g(f(x)) are equal to x, so d) is a pair of inverse functions.
Therefore, the pairs of functions that are inverses of each other are b) f(x) = x + 10 and g(x) = 4x - 10, and d) f(x) = 6x^2 - 7 and g(x) = x + 7.