Given:
[tex]f(x)=\dfrac{8}{x}[/tex]
[tex]g(x)=\dfrac{8}{x}[/tex]
To find:
Whether f(x) and g(x) are inverse of each other by using that f(g(x)) = x and g(f(x)) = x.
Solution:
We know that, two function are inverse of each other if:
[tex]f(g(x))=x[/tex] and [tex]g(f(x))[/tex]
We have,
[tex]f(x)=\dfrac{8}{x}[/tex]
[tex]g(x)=\dfrac{8}{x}[/tex]
Now,
[tex]f(g(x))=f(\dfrac{8}{x})[/tex] [tex][\because g(x)=\dfrac{8}{x}][/tex]
[tex]f(g(x))=\dfrac{8}{\dfrac{8}{x}}[/tex] [tex][\because f(x)=\dfrac{8}{x}][/tex]
[tex]f(g(x))=8\times \dfrac{x}{8}[/tex]
[tex]f(g(x))=x[/tex]
Similarly,
[tex]g(f(x))=f(\dfrac{8}{x})[/tex] [tex][\because f(x)=\dfrac{8}{x}][/tex]
[tex]g(f(x))=\dfrac{8}{\dfrac{8}{x}}[/tex] [tex][\because g(x)=\dfrac{8}{x}][/tex]
[tex]g(f(x))=8\times \dfrac{x}{8}[/tex]
[tex]g(f(x))=x[/tex]
Since, [tex]f(g(x))=x[/tex] and [tex]g(f(x))[/tex], therefore, f(x) and g(x) are inverse of each other.
