The Concept of an Inverse Function

Core Idea: An inverse function reverses what the original function does.

If $$f$$ takes $x$ to $y$, then $$f^{-1}$$ takes y back to $x$.

Formal Definition:
Functions f and g are inverses if:

$$f(g(x))=x$$ for all x in domain of g

$$g(f(x))=x$$ for all x in domain of f

Notation: $$f^{-1}(x)$$ (read “f inverse”)
⚠️ Warning: $$f^{-1}(x)\neq\frac{1}{f(x)}$$ (that’s the reciprocal!)

The Horizontal Line Test

A function has an inverse function (is invertible) if and only if it is one-to-one (1-1).

One-to-one: Each y-value corresponds to exactly ONE x-value.

Graphical Test: If EVERY horizontal line intersects the graph at most once, the function is 1-1 and has an inverse function.

Examples:

• $$f(x)=x^2$$ fails (horizontal line at y=4 hits at x=2 and x=-2)

• $$f(x)=x^3$$ passes (any horizontal line hits exactly once)

Finding Inverses Algebraically

Step-by-Step Process:

1. Replace f(x) with $y$

2. Swap x and y (this is the inverse relationship)

3. Solve for y

4. Replace y with $$f^{-1}(x)$$

5. Verify by checking $$f(f^{-1}(x))=x$$ and $$f^{-1}(f(x))=x$$

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

1. y=2x+3

2. Swap: x=2y+3

3. Solve: x−3=2y ⇒ $$y=\frac{x-3}{2}$$​

4. $$f^{-1}(x)=\frac{x-3}{2}$$​

5. Check: $$f(f^{-1}(x))=2\left(\frac{x-3}{2}\right)+3=x$$

Domain and Range of Inverses

Key Relationship:
Domain of $$f^{-1}$$ = Range of f
Range of $$f^{-1}$$ = Domain of f

Example : $$f(x)=\sqrt{x}$$​ has domain [0,∞), range [0,∞)
Its inverse $$f^{-1}(x)=x^2$$ (with restricted domain!) has domain [0,∞), range [0,∞)
Wait—this seems symmetric. Actually, $$f^{-1}(x)=x^2$$ with x≥0

Important: Sometimes we must restrict the domain of the original function to make it invertible.

Graphical Relationship: The Line y = x

The graph of f−1f−1 is the reflection of $f$‘s graph across the line y=x.

Why? Swapping x and y geometrically reflects across y=x.

Example : $$f(x)=2^x$$ and $$f^{-1}(x)=\log_{2}x$$ are reflections across y=x.

Finding Inverses of Non-One-to-One Functions

For functions that fail the horizontal line test, we can restrict the domain to make them invertible.

Example 4: $$f(x)=x^2$$ (not 1-1 over all reals)

• Restrict to x≥0: Inverse is $$f^{-1}(x)=\sqrt{x}$$​

• Restrict to x≤0: Inverse is $$f^{-1}(x)=-\sqrt{x}$$​

The inverse depends on which branch you choose!​