Finding the inverse of a linear function
Find the inverse of a linear function in Grade 10 algebra. Swap x and y, solve for y, and verify inverse correctness by confirming f(f⁻¹(x))=x and f⁻¹(f(x))=x.
Key Concepts
To find an equation for the inverse of a function like $y = 2x + 6$, first interchange the variables (swap $x$ and $y$). Then, solve the resulting equation for $y$ to get the inverse function's formula.
Find the inverse of $y = 5x + 10$. Swap variables to get $x = 5y + 10$. Solve for y: $x 10 = 5y$, so $y = \frac{1}{5}x 2$.: The inverse of $y = 3x + 9$ is found by swapping to $x = 3y + 9$. Solving gives $x 9 = 3y$, so $y = \frac{1}{3}x + 3$.
Think of this as a two step mission to uncover a function's secret identity. First, you pull the ultimate switcheroo: $x$ becomes $y$ and $y$ becomes $x$. Then, you unleash your algebra skills to solve for the new $y$, isolating it on one side. Mission accomplished! You have successfully found the formula for the inverse function.
Common Questions
How do you find the inverse of a linear function?
Replace f(x) with y, swap x and y, then solve the new equation for y. The result is f⁻¹(x). For y = 2x + 6: swap to get x = 2y + 6, solve: y = (x - 6)/2 = f⁻¹(x).
How do you verify that two functions are inverses of each other?
Compute f(f⁻¹(x)) and f⁻¹(f(x)). Both must equal x. If either composition does not simplify to x, they are not inverses.
What does the graph of an inverse function look like?
The graph of f⁻¹ is the reflection of f across the line y = x. Every point (a, b) on f corresponds to (b, a) on f⁻¹.