Graphing Inverse Functions by Reflection
The graph of an inverse function f⁻¹(x) is the reflection of f(x) across the line y = x — not across the x-axis or y-axis — a key transformation concept in enVision Algebra 1 Chapter 10 for Grade 11. Reflecting swaps x and y coordinates: if (1, 3) is on f(x) = 2x + 1, then (3, 1) is on f⁻¹(x). For f(x) = x² (domain x ≥ 0) with point (2, 4), the reflected point on f⁻¹(x) = √x is (4, 2). If (0, 5) is on the original function, then (5, 0) is on its inverse. This reflection property provides a visual method for graphing inverses without finding the algebraic formula.
Key Concepts
The graph of an inverse function $f^{ 1}(x)$ is the reflection of the original function $f(x)$ across the line $y = x$, not across the x axis or y axis.
Common Questions
How do you graph the inverse of a function by reflection?
Reflect every point (x, y) on f(x) to the point (y, x) across the line y = x. The resulting set of points is the graph of f⁻¹(x).
If (1, 3) is on f(x), what point is on f⁻¹(x)?
(3, 1). Reflection across y = x swaps the x and y coordinates of every point.
What is the reflected point of (2, 4) on f(x) = x²?
The reflected point is (4, 2), which lies on f⁻¹(x) = √x. Check: √4 = 2.
Why is the inverse reflected across y = x and not the x or y axis?
The inverse swaps input and output. Reflecting across y = x geometrically swaps x and y coordinates, which is exactly what inverse functions do algebraically.
Does every function have an inverse that is also a function?
No. Only one-to-one functions (passing the horizontal line test) have inverses that are also functions. For f(x) = x², we restrict the domain to x ≥ 0 to ensure the inverse √x is a function.