Question

Problem 1: Verify whether these two functions are inverses: f(x) = (x + 5)/(2x+1) and g(x) = (5-x)/(2x-1) What is g(f(x))? Where do i start? Do I need to find the inverses first?

Answer 1

Answer:

f(x) and g(x) are inverse functions because:

f(g(x)) = x and g(f(x)) = x

Explanation:

Two functions f(x) and g(x) are inverses if g(f(x)) = x and f(g(x)) = x.

So, f(x) and g(x) are equal to:

$\begin{gathered} f(x)=\frac{x+5}{2x+1} \\ g(x)=\frac{5-x}{2x-1} \end{gathered}$

Then, to find g(f(x)), we need to replace x by f(x) on the equation of g(x), so we get:

$\begin{gathered} g(f(x))=\frac{5-f(x)}{2f(x)-1} \\ g(f(x))=\frac{5-\frac{x+5}{2x+1}}{2(\frac{x+5}{2x+1})-1} \\ g(f(x))=\frac{\frac{5(2x+1)-(x+5)}{2x+1}}{\frac{2(x+5)-1(2x+1)}{2x+1}} \\ g(f(x))=\frac{5(2x+1)-(x+5)}{2(x+5)-(2x+1)_{}} \\ g(f(x))=\frac{10x+5-x-5}{2x+10-2x-1} \\ g(f(x))=\frac{9x}{9}=x \end{gathered}$

Now, we need to verify that f(g(x)) is also equal to x, so:

$\begin{gathered} f(g(x))=\frac{g(x)+5}{2g(x)+1} \\ f(g(x))=\frac{\frac{5-x}{2x-1}+5}{2(\frac{5-x}{2x+1})+1} \\ f(g(x))=\frac{\frac{5-x+5(2x-1)}{2x-1}}{\frac{2(5-x)+1(2x-1)}{2x-1}} \\ f(g(x))=\frac{(5-x)+5(2x-1)}{2(5-x)+(2x-1)} \\ f(g(x))=\frac{5-x+10x-5}{10-2x+2x-1} \\ f(g(x))=\frac{9x}{9}=x \end{gathered}$

Since g(f(x)) and f(g(x)) are equal to x, we can say that f(x) ad g(x) are inverse functions.