Inverse Function Calculator with Steps

Linear Function Inverse Calculator

Enter the slope (m) and y-intercept (b) of your linear function f(x) = mx + b to find its inverse function f⁻¹(x) and the step-by-step derivation.

Inverse Function Calculation Steps:

f(x) = " + m + "x + " + b + "

Step 1: Replace f(x) with y."; output += "y = " + m + "x + " + b + "

Step 2: Swap x and y."; output += "x = " + m + "y + " + b + "

Step 3: Isolate the term containing y."; output += "Subtract " + b + " from both sides:"; output += "x – " + b + " = " + m + "y

Step 4: Solve for y."; output += "Divide both sides by " + m + ":"; var numerator = "x – " + b; if (b < 0) { // Handle negative b for cleaner display numerator = "x + " + Math.abs(b); } output += "y = (" + numerator + ") / " + m + "

Understanding Inverse Functions and How to Calculate Them

An inverse function, often denoted as f⁻¹(x), essentially "undoes" what the original function f(x) does. If a function takes an input x and produces an output y, its inverse function takes that y as an input and returns the original x. Think of it like putting on your shoes (the function) and then taking them off (the inverse function) – you end up back where you started.

Key Concepts of Inverse Functions

• One-to-One Functions: For an inverse function to exist, the original function must be "one-to-one." This means that every unique input x maps to a unique output y, and no two different x values produce the same y value. Graphically, this means the function passes the horizontal line test (any horizontal line intersects the graph at most once).
• Domain and Range Swap: The domain of the original function becomes the range of its inverse, and the range of the original function becomes the domain of its inverse.
• Symmetry: The graph of a function and its inverse are symmetric with respect to the line y = x.

General Steps to Find an Inverse Function

While the specific algebraic manipulations vary depending on the function type, the general process for finding an inverse function is as follows:

1. Replace f(x) with y: This makes the equation easier to manipulate.
2. Swap x and y: This is the crucial step that conceptually "inverts" the relationship between input and output.
3. Solve the new equation for y: Use algebraic techniques to isolate y on one side of the equation.
4. Replace y with f⁻¹(x): Once y is isolated, it represents the inverse function.

Finding the Inverse of a Linear Function (f(x) = mx + b)

Linear functions are excellent examples for understanding inverse functions because they are always one-to-one (unless the slope m is zero). Let's walk through the steps with an example.

Example: Find the inverse of f(x) = 2x + 3

1. Replace f(x) with y:
   y = 2x + 3
2. Swap x and y:
   x = 2y + 3
3. Solve for y:
   • Subtract 3 from both sides:
     x - 3 = 2y
   • Divide both sides by 2:
     y = (x - 3) / 2
4. Replace y with f⁻¹(x):
   f⁻¹(x) = (x - 3) / 2

This means if f(x) takes an input, multiplies it by 2, and adds 3, its inverse f⁻¹(x) will take an input, subtract 3, and then divide by 2, effectively reversing the operations.

When an Inverse Function Does Not Exist

As mentioned, a function must be one-to-one for its inverse to exist. A common example of a function that is NOT one-to-one is f(x) = x². For instance, f(2) = 4 and f(-2) = 4. Since two different inputs (2 and -2) produce the same output (4), this function fails the horizontal line test and does not have a unique inverse over its entire domain. If we restrict the domain (e.g., to x ≥ 0), then an inverse (f⁻¹(x) = √x) can be found.

The calculator above specifically handles linear functions, which are always one-to-one (unless the slope is zero, in which case it's a horizontal line and has no inverse).