Given:
The two functions are
[tex]f(x)=5x+1[/tex]
[tex]g(x)=\dfrac{x}{5}-\dfrac{1}{5}[/tex]
To find:
The presentation which establishes that the functions are inverse functions.
Solution:
Two functions f(x) and g(x) are inverse of each other, if
[tex]f(g(x))=g(f(x))=x[/tex]
We have,
[tex]f(x)=5x+1[/tex]
[tex]g(x)=\dfrac{x}{5}-\dfrac{1}{5}[/tex]
Now,
[tex]g(f(x))=\dfrac{5x+1}{5}-\dfrac{1}{5}[/tex]
[tex]g(f(x))=\dfrac{5x}{5}+\dfrac{1}{5}-\dfrac{1}{5}[/tex]
[tex]g(f(x))=x[/tex]
Similarly,
[tex]f(g(x))=5\left(\dfrac{x}{5}-\dfrac{1}{5}\right)+1[/tex]
[tex]f(g(x))=x-1+1[/tex]
[tex]f(g(x))=x[/tex]
Since, [tex]f(g(x))=g(f(x))=x[/tex], therefore, the given functions are inverse of each other.
Hence, the correct option is C.
