Inverse of a Function

Activity 1: Real-World Analogy (सादृश्य)

Consider two airlines flying between Pokhara and Kathmandu:

Scenario: Yeti Air carries a person from Pokhara to Kathmandu. Buddha Air carries a person from Kathmandu to Pokhara. Buddha Air acts opposite to Yeti Air, undoing whatever Yeti Air does.

Observation:

Yeti Air carries a person from Pokhara to Kathmandu
Buddha Air carries a person from Kathmandu to Pokhara
Buddha Air acts opposite to Yeti Air
Whatever work is done by Yeti Air is undone by Buddha Air

$$(\text{Buddha} \circ \text{Yeti})(\text{Pokhara}) = \text{Buddha}(\text{Yeti}(\text{Pokhara}))$$ $$= \text{Buddha}(\text{Kathmandu}) = \text{Pokhara}$$

Conclusion: $(\text{Buddha} \circ \text{Yeti})(\text{Pokhara}) = \text{Pokhara}$ — This is the Identity Function. Buddha Air is the inverse of Yeti Air.

Activity 2: Function Machine

Consider two function machines with input $x = 3$:

Setup: Machine $f$ transforms 3 to 4. Machine $g$ transforms 4 back to 3. The work done by $f$ is reversed by $g$.

Observation:

$f(3) = 4$
$g(4) = 3$
The action of $f$ is undone/reversed by $g$

$$(g \circ f)(3) = g(f(3)) = g(4) = 3$$

Conclusion: $(g \circ f)(3) = 3$ — The Identity Function is confirmed. Function $g$ is the inverse of function $f$.

D

Formal Definition

Inverse Function

Definition: Let $f: A \to B$ be a function from set $A$ to set $B$. Let $g: B \to A$ be a function defined from set $B$ to $A$. The function $g$ is said to be the inverse of $f$ (or vice-versa) if: $$f \circ g(x) = g \circ f(x) = x \quad \text{for all } x \in \text{their domains}$$ The inverse is denoted as: $g = f^{-1}$ or $f = g^{-1}$

📌 Important Note: For the existence of an inverse, the function must be one-to-one and onto (bijective). This ensures that every element in the codomain has exactly one pre-image in the domain.

E

Worked Examples

Example 1: Finding Inverse of a Relation

Find $f^{-1}$ if $f = \{(a,1),\,(b,2),\,(c,3)\}$

Solution:

To find the inverse, swap every ordered pair $(x, y) \to (y, x)$:

$$f^{-1} = \{(1,a),\,(2,b),\,(3,c)\}$$

The original function maps {$a, b, c$} → {1, 2, 3}. The inverse function reverses this mapping, sending {1, 2, 3} → {$a, b, c$}.

Example 2: Finding Inverse of an Algebraic Function

Find $f^{-1}(x)$ if $f(x) = 2x + 3$

Solution:

Step I: Let $y = f(x)$ $$y = 2x + 3$$

Step II: Interchange variables $x$ and $y$

Replace every $x$ with $y$ and every $y$ with $x$:

$$x = 2y + 3$$

Step III: Solve for $y$ $$x - 3 = 2y$$ $$y = \frac{x - 3}{2}$$

Step IV: Write the inverse $$f^{-1}(x) = \frac{x-3}{2}$$

Verification (Checking):

We verify by computing $f \circ f^{-1}(x)$:

$$f(f^{-1}(x)) = f\left(\frac{x-3}{2}\right) = 2 \cdot \frac{x-3}{2} + 3 = (x - 3) + 3 = x$$

✓ Verified: Since $f(f^{-1}(x)) = x$, we have confirmed that $f^{-1}(x) = \dfrac{x-3}{2}$ is the correct inverse function. The composition gives us the identity function.

Why Does This Method Work?

Mathematical Reasoning:

If $y = f(x)$, then interchanging $x$ and $y$ gives us $x = f(y)$.

When we apply $f^{-1}$ to both sides:

$$f^{-1}(x) = f^{-1}(f(y)) = (f^{-1} \circ f)(y) = y$$

This works because the composite of a function and its inverse is the identity function, so $f^{-1}(f(y)) = y$.

K

Key Takeaways

Inverse Concept: An inverse function reverses or "undoes" the action of the original function.
Existence Condition: Inverses exist only for bijective functions (one-to-one and onto).
Composition Property: $f(f^{-1}(x)) = f^{-1}(f(x)) = x$ (identity property).
Finding Inverses: To find $f^{-1}(x)$, replace $f(x)$ with $y$, swap $x$ and $y$, then solve for $y$.
Verification: Always verify your inverse by checking that the composition equals the identity function.

Inverse of a Function

Inverse Functions

Understanding Inverse Functions

Formal Definition

Inverse Function

Worked Examples

Solution:

Solution:

Verification (Checking):

Why Does This Method Work?

Key Takeaways