Certainly! To find the inverse of the function
[tex]\[ f(n) = \frac{2}{n+1} \][/tex]
we will follow a series of steps to express the given equation in terms of its inverse.
1. Introduce a new variable: Let
[tex]\[ y = f(n) = \frac{2}{n+1} \][/tex]
2. Express the function in terms of \( y \): We need to rearrange the equation
[tex]\[ y = \frac{2}{n+1} \][/tex]
to isolate \( n \).
3. Multiply both sides by \( n + 1 \) to clear the fraction:
[tex]\[ y(n + 1) = 2 \][/tex]
4. Distribute \( y \) on the left-hand side:
[tex]\[ yn + y = 2 \][/tex]
5. Isolate \( n \): Subtract \( y \) from both sides to get:
[tex]\[ yn = 2 - y \][/tex]
6. Solve for \( n \): Divide both sides by \( y \):
[tex]\[ n = \frac{2 - y}{y} \][/tex]
Thus, the inverse function, expressed as \( n \) in terms of \( y \), is:
[tex]\[
f^{-1}(y) = \frac{2 - y}{y}
\][/tex]
So the inverse of the function \( \frac{2}{n+1} \) is:
[tex]\[
\frac{2 - y}{y}
\][/tex]
