Inverse of a Function Game

Inverse of a Function Game
This game will require you to find the inverse of a function. Scroll down for a detailed explanation.
 

 

How to Play the Inverse of a Function Game
Analyze the Target: Look at the given function f(x).
Identify the Operations: Determine what is happening to x and in what order.
Reverse Everything: To find f^-1(x), you must apply the inverse operations in the reverse order.
Select the Match: Click one of the four buttons.
Green: Correct. You earn 500 points and a new problem loads.
Red: Incorrect. The card shakes, and that option is disabled. Try again.

The “Inverse” Strategy
Linear Functions (ax + b)
The Path: Multiply by a, then add b.
The Inverse: Subtract b, then divide by a.
Look for: (x - b) / a.

Fractional Functions
(\(\frac{1}{d}x - b\))
The Path: Divide by d, then subtract b.
The Inverse: Add b, then multiply the whole thing by d.
Look for: d(x + b).

Quadratic Functions
(x² + k)
The Path: Square x, then add k.
The Inverse: Subtract k, then take the square root.
Look for: \(\sqrt{x - k}\).

Vertex Form
((x - h)²)
The Path: Subtract h, then square the result.
The Inverse: Take the square root, then add h.
Look for: \(\sqrt{x} + h\).

How to find the inverse of a function?
To find the inverse of a function, you are essentially creating a “reverse map.” If the original function f(x) takes an input and turns it into y, the inverse function f^-1(x) takes that y and brings you back to the original input.
Mathematically, the goal is to undo every operation in the exact opposite order they were applied.

The 4-Step Algebraic Method
This is the most reliable way to find the inverse for almost any algebraic function.

1. Replace f(x) with y
   This makes the equation easier to manipulate.
   Example: f(x) = 3x + 5 → y = 3x + 5
   

2. Swap x and y
   This is the “Inverse” step. Since an inverse switches inputs and outputs, you literally switch the variables in the equation.
   Example: x = 3y + 5
   

3. Solve for y
   Use algebra to isolate y on one side of the equation. This is where you perform the “reverse operations."
   Subtract 5 from both sides: x - 5 = 3y
   Divide both sides by 3: \(\frac{x - 5}{3} = y\)
   

4. Replace y with f^-1(x)
   This is the formal notation for the inverse function.
   Result:
   \(f^{-1}(x) = \frac{x - 5}{3}\)
   

This video gives a clear, step-by-step approach to find the inverse of a function.

• Show Video

 

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