Thinking inverse means reciprocal H
"The Reciprocal Trap"
The Mistake in Action
Given $f(x) = 2x + 3$. Find $f^{-1}(x)$.
Wrong: $f^{-1}(x) = \frac{1}{2x + 3}$
Why It Happens
Students confuse $f^{-1}(x)$ (inverse function) with $\frac{1}{f(x)}$ (reciprocal) because of the negative exponent notation.
The Fix
$f^{-1}(x)$ is NOT $\frac{1}{f(x)}$!
The inverse function reverses what $f$ does. If $f$ takes 1 to 5, then $f^{-1}$ takes 5 back to 1.
To find $f^{-1}(x)$:
- Let $y = 2x + 3$
- Swap: $x = 2y + 3$
- Rearrange: $x - 3 = 2y$, so $y = \frac{x - 3}{2}$
- Therefore: $f^{-1}(x) = \frac{x - 3}{2}$
Check: $f(1) = 5$ and $f^{-1}(5) = \frac{5-3}{2} = 1$ ✓
Spot the Mistake
Can you identify where this student went wrong?
Find $f^{-1}(x)$ where $f(x) = 2x + 3$
$f^{-1}(x) = \frac{1}{2x + 3}$
Click on the line that contains the error.
Related Topics
Learn more about the underlying maths: