Show that f and g are inverse functions (a) algebraically and(b) graphically. f(x)=7 x+1, g(x)

step 1 We have the functions f of x equals 7x plus 1 and g of x equals x minus 1 over 7. First we want

Show that f and g are inverse functions (a) algebraically...

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Key Concepts

Inverse Functions

Inverse functions are pairs of functions where one function 'undoes' the effect of the other. This means that if one function (f) is applied to an input and then its inverse (g) is applied to the result, the original input is obtained, and vice versa. Formally, this is represented as f(g(x)) = x and g(f(x)) = x for every x in their respective domains.

Algebraic Verification

Algebraic verification of inverse functions involves composing the functions in both orders (f ? g and g ? f) and simplifying the expressions to check if the results equal the identity function. This method provides a rigorous proof that the functions indeed undo each other’s actions through algebraic manipulation.

Graphical Verification

Graphical verification of inverse functions is based on the property that the graph of an inverse function is a reflection of the original function’s graph across the line y = x. By plotting both functions on the same set of axes, one can visually inspect whether the functions display the required symmetry, confirming that they are inverses.

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