Q. What is the inverse of the function \(f(x)=2x+1\)

Problem: Find the inverse of the function \( f(x) = 2x + 1 \)

Step 1 — Rewrite using \(y\)

Write the function as
\[
y = 2x + 1,
\]
so we can solve algebraically for the input in terms of the output.

Step 2 — Swap \(x\) and \(y\)

Interchange \(x\) and \(y\) to reflect the inverse relationship:
\[
x = 2y + 1.
\]

Step 3 — Solve for \(y\)

Subtract 1 from both sides:
\[
x – 1 = 2y.
\]
Now divide both sides by 2:
\[
y = \frac{x – 1}{2}.
\]

Step 4 — Write the inverse function

Replace \(y\) with \(f^{-1}(x)\):
\[
f^{-1}(x) = \frac{x – 1}{2}.
\]

Step 5 — Verify by composition

Compute \(f(f^{-1}(x))\):
\[
f\bigl(f^{-1}(x)\bigr)=2\left(\frac{x-1}{2}\right)+1 = x-1+1 = x.
\]
Compute \(f^{-1}(f(x))\):
\[
f^{-1}\bigl(f(x)\bigr)=\frac{(2x+1)-1}{2}=\frac{2x}{2}=x.
\]
Both compositions return \(x\), confirming the inverse is correct.

Answer

\[
f^{-1}(x)=\frac{x-1}{2}
\]