Answer:
The only pair of functions that are inverses of each other are the ones for option D.
Step-by-step explanation:
Two functions, f(x) and g(x), are inverses if and only if:
f( g(x) ) = x
g( f(x) ) = x
So we need to check that with all the given options.
A)
[tex]f(x) = \frac{x}{7} + 10 \\g(x) = 7*x - 10\\[/tex]
then:
[tex]f(g(x)) = \frac{7*x + 10}{7} -10 = x + \frac{10}{7} - 10[/tex]
This is clearly different than x, so f(x) and g(x) are not inverses.
B)
[tex]f(x) = \sqrt[3]{11*x} \\g(x) = (\frac{x}{11} )^3[/tex]
Then:
[tex]f(g(x)) = \sqrt[3]{11*(\frac{x}{11})^3 } = \sqrt[3]{\frac{x^3}{11^2} } = \frac{x}{11^{2/3}}[/tex]
This is different than x, so f(x) and g(x) are not inverses.
C)
[tex]f(x) = \frac{7}{x} -2 \\g(x) = \frac{x + 2}{7}[/tex]
Then:
[tex]f(g(x)) = \frac{7}{\frac{x + 2}{7} } - 2 = \frac{7*7}{x + 2} - 2[/tex]
Obviously, this is different than x, so f(x) and g(x) are not inverses.
D)
[tex]f(x) = 9*x - 6\\g(x) = \frac{x + 6}{9}[/tex]
Then:
[tex]f(g(x)) = 9*\frac{x + 6}{9} - 6 = x + 6 - 6 = x\\g(f(x)) = \frac{(9*x - 6) + 6}{9} = x[/tex]
In this case we can conclude that f(x) and g(x) are inverses of each other.
