Answer:
The pair which are inverse of each other are:
Option: C
[tex]f(x)=11x-4\ and\ g(x)=\dfrac{x+4}{11}[/tex]
Step-by-step explanation:
Two functions f(x) and g(x) are said to be inverse of each other if:
fog(x)=gof(x)=x
i.e. the composition of two functions give identity no matter what the order is.
A)
[tex]f(x)=\dfrac{x}{12}+15[/tex] and [tex]g(x)=12x-15[/tex]
Now we calculate fog(x):
[tex]fog(x)=f(g(x))\\\\fog(x)=f(12x-15)\\\\fog(x)=\dfrac{12x-15}{12}+15\\\\fog(x)=x-\dfrac{15}{12}+15\\\\fog(x)=x-\dfrac{55}{4}\neq x[/tex]
Hence, option: A is incorrect.
B)
[tex]f(x)=\dfrac{3}{x}-10\ and\ g(x)=\dfrac{x+10}{2}[/tex]
Now we calculate fog(x):
[tex]fog(x)=f(g(x))\\\\fog(x)=f(\dfrac{x+10}{3})\\\\fog(x)=\dfrac{3}{\dfrac{x+10}{3}}-10\\\\fog(x)=\dfrac{9}{x+10}-10\neq x[/tex]
Hence, option: B is incorrect.
D)
[tex]f(x)=9+\sqrt[3]{x}\ and\ g(x)=9-x^3[/tex]
Now we calculate fog(x):
[tex]fog(x)=f(g(x))\\\\fog(x)=f(9-x^3)[/tex]
[tex]fog(x)=9+\sqrt[3]{9-x^3}\neq x[/tex]
Hence, option: D is incorrect.
C)
[tex]f(x)=11x-4\ and\ g(x)=\dfrac{x+4}{11}[/tex]
Now we calculate fog(x):
[tex]fog(x)=f(g(x))\\\\fog(x)=f(\dfrac{x+4}{11})[/tex]
[tex]fog(x)=11\times (\dfrac{x+4}{11})-4\\\\fog(x)=x+4-4\\\\fog(x)=x[/tex]
Similarly,
[tex]gof(x)=g(f(x))\\\\gof(x)=g(11x-4)\\\\gof(x)=\dfrac{11x-4+4}{11}\\\\gof(x)=\dfrac{11x}{11}\\\\gof(x)=x[/tex]
Hence, option: C is the correct option.
