Slope of an Inverse Linear Function: Reciprocal, Not Negative Reciprocal
In Grade 9 Algebra 1 (California Reveal Math, Unit 4), this skill teaches that the slope of an inverse linear function equals the reciprocal of the original slope — not the negative reciprocal. For f(x) = mx + b, the inverse has slope 1/m. For example, if f(x) = 3x + 6, the inverse slope is 1/3, not -1/3. Students learn to distinguish this from perpendicular lines, which use the negative reciprocal -1/m. This critical distinction prevents the most common error students make when finding inverse functions algebraically.
Key Concepts
If $f(x) = mx + b$ (with $m \neq 0$), then its inverse function is:.
$$f^{ 1}(x) = \frac{1}{m}x \frac{b}{m}$$.
Common Questions
What is the slope of the inverse of f(x) = 3x + 6?
The inverse has slope 1/3 (the reciprocal of 3). The full inverse is f⁻¹(x) = (1/3)x - 2. A common error is writing -1/3, which is the perpendicular slope, not the inverse slope.
How is the slope of an inverse function different from a perpendicular line?
Inverse function: slope is 1/m (reciprocal). Perpendicular line: slope is -1/m (negative reciprocal). For slope 2, the inverse slope is 1/2 while the perpendicular slope is -1/2.
Why does the inverse of f(x) = mx + b have slope 1/m?
When you swap x and y and solve for y algebraically, dividing by m gives slope 1/m. The inverse undoes multiplication by m by dividing by m instead.
What is the slope of the inverse of f(x) = -4x + 1?
The original slope is -4, so the inverse slope is 1/(-4) = -1/4. The negative stays with the reciprocal. The full inverse is f⁻¹(x) = -1/4 x + 1/4.
What is the most common error students make with inverse function slopes?
Confusing the inverse slope (reciprocal 1/m) with the perpendicular slope (negative reciprocal -1/m). For f(x) = 2x - 5, the correct inverse slope is +1/2, not -1/2.