If f(x) = (x + 3)/(4x - 5) and g(x) = (5x + 3)/(4x - 1), then show that fog(x) = x.

Understand the Problem

The question is asking to demonstrate that the composition of two functions, denoted as fog(x), is equal to x, given specific expressions for the functions f(x) and g(x). This involves substituting one function into the other and simplifying to show they equal x.

Answer

The composition $ fog(x) = x $.

Steps to Solve

Define the Functions We have two functions defined as follows:

( f(x) = \frac{x + 3}{4x - 5} )
( g(x) = \frac{5x + 3}{4x - 1} )

Compute ( fog(x) ) To find ( fog(x) ), we substitute ( g(x) ) into ( f(x) ): $$ fog(x) = f(g(x)) = f\left(\frac{5x + 3}{4x - 1}\right) $$
Substituting ( g(x) ) into ( f(x) ) Now substitute ( g(x) ) into ( f(x) ): $$ f\left(g(x)\right) = \frac{\left(\frac{5x + 3}{4x - 1}\right) + 3}{4\left(\frac{5x + 3}{4x - 1}\right) - 5} $$ Simplify the numerator:

The numerator becomes: $$ \frac{5x + 3 + 3(4x - 1)}{4x - 1} = \frac{5x + 3 + 12x - 3}{4x - 1} = \frac{17x}{4x - 1} $$

Simplifying the Denominator Now simplify the denominator:

The denominator becomes: $$ 4\left(\frac{5x + 3}{4x - 1}\right) - 5 = \frac{20x + 12}{4x - 1} - \frac{5(4x - 1)}{4x - 1} = \frac{20x + 12 - 20x + 5}{4x - 1} = \frac{17}{4x - 1} $$

Putting It All Together Now putting the numerator and denominator together: $$ fog(x) = \frac{\frac{17x}{4x - 1}}{\frac{17}{4x - 1}} = \frac{17x}{17} = x $$
Conclusion Thus, we have shown that ( fog(x) = x ).

The composition ( fog(x) ) is equal to ( x ).

More Information

This result indicates that the composition of the functions ( f(g(x)) ) returns the original input ( x ), implying that ( f ) and ( g ) are inverse functions of each other.

Tips

Misinterpreting Function Composition: Make sure to substitute ( g(x) ) into ( f(x) ) correctly and keep track of the order of operations.
Neglecting Simplification Steps: Ensure to fully simplify both the numerator and denominator before concluding.

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