If `\(f(x) = \sqrt{x-3}\)`, Then Verify `\((f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x\)`

        If `\f\left(x\right) = \sqrt{x - 3}` then verify `\left(fof^{-1}\right)\left(x\right) = \left(f^- 1of\right)\left(x\right) = x` 

In this post, we will verify the composition of a function f(x) = √(x − 3) with its inverse function and show that both (f∘f⁻¹)(x) and (f⁻¹∘f)(x) equal x.

Step 1: Find the inverse function f⁻¹(x)

Let y = f(x) = √(x − 3)

Swap x and y to find the inverse:

x = √(y − 3)

Square both sides:

x² = y − 3 → y = x² + 3

So, f⁻¹(x) = x² + 3

Step 2: Verify (f ∘ f⁻¹)(x)

(f ∘ f⁻¹)(x) = f(f⁻¹(x)) = f(x² + 3)

Substitute into f(x):

f(x² + 3) = √((x² + 3) − 3) = √(x²) = |x|

Since the domain of f⁻¹(x) is x ≥ 0, we have √(x²) = x

✅ Therefore, (f ∘ f⁻¹)(x) = x

Step 3: Verify (f⁻¹ ∘ f)(x)

(f⁻¹ ∘ f)(x) = f⁻¹(f(x)) = f⁻¹(√(x − 3))

Substitute into f⁻¹(x):

f⁻¹(√(x − 3)) = (√(x − 3))² + 3 = x − 3 + 3 = x

✅ Therefore, (f⁻¹ ∘ f)(x) = x

Conclusion

We have verified that the composition of a function and its inverse satisfies the property:

• (f ∘ f⁻¹)(x) = x
• (f⁻¹ ∘ f)(x) = x

This confirms the relationship between a function and its inverse for the given function f(x) = √(x − 3).

FAQs

1. What is the domain of f(x)?

The domain of f(x) = √(x − 3) is x ≥ 3, because the expression under the square root must be non-negative.

2. What is the range of f(x)?

The range of f(x) is y ≥ 0, because the square root function always gives non-negative outputs.

3. Why do we take only the positive square root in (f ∘ f⁻¹)(x)?

Because the range of f⁻¹(x) is x ≥ 0, so √(x²) = x, not −x.

4. Can all functions have an inverse?

No, only one-to-one functions (bijective functions) have inverses. Functions must be injective and surjective in their domain and range.

By following these steps, you can verify function-inverse compositions for other functions as well.

Explore more about GENERAL MATHEMATICS in Mathematics Notes & MCQs.