Inverse Function Point Transformer
x*x, 2*x + 1, Math.sqrt(x), Math.sin(x).
f(x) = " + funcString + ":";
resultDiv.innerHTML += "If x = " + xValue + ", then f(x) = " + yValue + ".";
resultDiv.innerHTML += "This means the point (" + xValue + ", " + yValue + ") is on the graph of f(x).";
resultDiv.innerHTML += "For its inverse function, f-1(x), the corresponding point is (" + yValue + ", " + xValue + ").";
resultDiv.innerHTML += "This demonstrates how points on a function's graph are reflected across the line y = x to form the graph of its inverse.";
} catch (e) {
resultDiv.innerHTML = "Error evaluating function: " + e.message + ". Please check your function syntax.";
}
}
Understanding Inverse Function Graphs
An inverse function, denoted as f-1(x), essentially "undoes" what the original function f(x) does. If f(a) = b, then it must be true that f-1(b) = a. This fundamental relationship has a very distinct and beautiful graphical interpretation.
The Reflection Property
The graph of an inverse function f-1(x) is a reflection of the graph of the original function f(x) across the line y = x. This means that if a point (a, b) lies on the graph of f(x), then the point (b, a) will lie on the graph of f-1(x).
Think of it this way: the x-coordinates and y-coordinates swap roles. The input of the original function becomes the output of the inverse, and vice-versa. When you swap the x and y coordinates of every point on a graph, the resulting set of points forms a mirror image across the line y = x.
How to Visualize the Reflection
- Draw the line
y = x: This is a straight line passing through the origin (0,0) with a slope of 1. It acts as the mirror. - Pick a point on
f(x): Let's say you have a point(3, 7)on the graph off(x). - Reflect the point: To find the corresponding point on
f-1(x), simply swap the coordinates. So,(7, 3)will be on the graph off-1(x). - Repeat for multiple points: By reflecting several key points, you can sketch the general shape of the inverse function's graph.
Using the Calculator
Our "Inverse Function Point Transformer" calculator helps you understand this reflection property by taking a specific x-value for your chosen function f(x). It then calculates the corresponding y-value, giving you a point (x, y) on f(x). Finally, it shows you the reflected point (y, x), which lies on the graph of f-1(x).
Examples:
Example 1: Linear Function
Let's use the function f(x) = 2*x + 3.
- If you input
X-value = 5:f(5) = 2*(5) + 3 = 10 + 3 = 13.- Point on
f(x): (5, 13). - Corresponding point on
f-1(x): (13, 5).
- If you input
X-value = -2:f(-2) = 2*(-2) + 3 = -4 + 3 = -1.- Point on
f(x): (-2, -1). - Corresponding point on
f-1(x): (-1, -2).
Example 2: Quadratic Function (with domain restriction for invertibility)
Consider f(x) = x*x (for x ≥ 0, as quadratic functions are not invertible over their entire domain).
- If you input
X-value = 4:f(4) = 4*4 = 16.- Point on
f(x): (4, 16). - Corresponding point on
f-1(x): (16, 4).
- If you input
X-value = 0:f(0) = 0*0 = 0.- Point on
f(x): (0, 0). - Corresponding point on
f-1(x): (0, 0). (The origin is on the line y=x, so it reflects onto itself).
Example 3: Square Root Function
Consider f(x) = Math.sqrt(x).
- If you input
X-value = 9:f(9) = Math.sqrt(9) = 3.- Point on
f(x): (9, 3). - Corresponding point on
f-1(x): (3, 9).
- If you input
X-value = 0.25:f(0.25) = Math.sqrt(0.25) = 0.5.- Point on
f(x): (0.25, 0.5). - Corresponding point on
f-1(x): (0.5, 0.25).
By experimenting with different functions and x-values, you can gain a deeper intuition for how inverse functions are graphically related to their original functions through this elegant reflection across the line y = x.