Four-Step Algebraic Method for Finding Inverses of Linear Functions
The four-step algebraic method for finding inverses of linear functions is taught in Grade 9 Algebra 1 (California Reveal Math, Unit 4). The steps are: (1) replace f(x) with y, (2) swap x and y, (3) solve for y, (4) replace y with f⁻¹(x). For f(x) = 2x + 6, swapping gives x = 2y + 6, solving yields y = (x-6)/2, so f⁻¹(x) = (x-6)/2. This systematic approach works for all linear functions with nonzero slope.
Key Concepts
To find the inverse of a linear function $f(x)$, follow these four steps:.
1. Replace $f(x)$ with $y$ 2. Swap $x$ and $y$ 3. Solve for $y$ 4. Replace $y$ with $f^{ 1}(x)$.
Common Questions
What are the four steps to find the inverse of a linear function?
Step 1: Replace f(x) with y. Step 2: Swap x and y. Step 3: Solve for y using algebra. Step 4: Replace y with f⁻¹(x). Every linear function with nonzero slope follows these same four steps.
How do you find the inverse of f(x) = 2x + 6?
Step 1: y = 2x + 6. Step 2: x = 2y + 6. Step 3: x - 6 = 2y, so y = (x-6)/2. Step 4: f⁻¹(x) = (x-6)/2.
How do you find the inverse of f(x) = (1/4)x - 2?
Step 1: y = (1/4)x - 2. Step 2: x = (1/4)y - 2. Step 3: x + 2 = (1/4)y, so y = 4(x+2). Step 4: f⁻¹(x) = 4x + 8.
Why is swapping x and y the key step in finding an inverse?
Swapping reflects the idea that an inverse reverses input and output. The original function maps x to f(x); the inverse maps f(x) back to x. After swapping, you solve for y to express the inverse explicitly.
Does this method work for all linear functions?
Yes, for all linear functions f(x) = mx + b where m is not zero. If m = 0, the function is constant and has no inverse, because every input maps to the same output.