Answer to Question #134546 in Calculus for benjamin

Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = x2 - 3 and g(x) = square root of quantity three plus x

Answer 1

$f(x) = x^2 - 3, g(x) = \sqrt{3 + x}\\ f(g(x)) \hspace{0.1cm} \textsf{means}\hspace{0.1cm}\textit{ f of g},\\ \textsf{i.e insert the function}\hspace{0.1cm} g\\\textsf{which is expressed as a function of }\hspace{0.1cm} x\\ \textsf{in the}\hspace{0.1cm} x \hspace{0.1cm}\textsf{in function} \hspace{0.1cm}f. \\ f(g(x)) = (\sqrt{3 + x})^2 - 3\\ f(g(x)) = 3 + x - 3 = x\\ \textsf{similarly}\hspace{0.1cm}g(f(x)) = \sqrt{3 + (x^2 - 3)} = \sqrt{3 + x^2 - 3} = \sqrt{x^2} = x\\ \therefore \textsf{Since}\hspace{0.1cm} f(g(x)) = g(f(x)) = x,\\ \textsf{functions} \hspace{0.1cm}f \hspace{0.1cm}\textsf{and} \hspace{0.1cm}g\hspace{0.1cm}\textsf{are inverses of one another.}$