Mathematics 10 — Exercise 4.1 · Q8 (i)

Full question

Verify f(f⁻¹(x))=f⁻¹(f(x))=x on the valid domains.

Step-by-step solution

1. Write the output as y. To find the inverse, solve this equation for the original input x.

   $$y=x-6$$

2. Subtract the constant from both sides.

   $$y+6=x$$

3. Divide both sides by the input coefficient.

   $$x=\frac{1}{1}\times \left(y+6\right)$$

4. Interchange the names x and y: the old output is now the inverse input.

   $$\mathrm{f⁻¹}\left(x\right)=x+6$$

5. The inverse domain is the original range. Keep any exclusions or nonnegative-input restriction.

   $$\mathrm{ℝ}$$

6. Compute f(f⁻¹(x)).

7. Composition means apply the inner function first, then use its entire output as the input of the outer function.

   $$x+6$$

8. Replace every x of the outer function by this full bracketed inner expression.

   $$-6+x+6$$

9. Substitute the function expressions, keeping the brackets.

   $$-6+x+6$$

10. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

    $$-6+x+6$$

11. Write EVERY distributed product before collecting terms.

    $$-6+6+x$$

12. Calculate this individual product, including its sign and powers.

    $$-6=-6$$

13. Calculate this individual product, including its sign and powers.

    $$6=6$$

14. Calculate this individual product, including its sign and powers.

    $$x=x$$

15. These terms have the same variable part. Add ONLY their coefficients.

    $$-6+6=0$$

16. These terms have the same variable part. Add ONLY their coefficients.

    $$1=1$$

17. Put the simplified groups together.

    $$0+x$$

18. Collect terms with the same power and simplify their coefficients.

    $$x$$

19. Compute f⁻¹(f(x)) independently.

20. Composition means apply the inner function first, then use its entire output as the input of the outer function.

    $$x-6$$

21. Replace every x of the outer function by this full bracketed inner expression.

    $$6+x-6$$

22. Substitute the function expressions, keeping the brackets.

    $$6+x-6$$

23. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

    $$6+x-6$$

24. Write EVERY distributed product before collecting terms.

    $$6-6+x$$

25. Calculate this individual product, including its sign and powers.

    $$6=6$$

26. Calculate this individual product, including its sign and powers.

    $$-6=-6$$

27. Calculate this individual product, including its sign and powers.

    $$x=x$$

28. These terms have the same variable part. Add ONLY their coefficients.

    $$6-6=0$$

29. These terms have the same variable part. Add ONLY their coefficients.

    $$1=1$$

30. Put the simplified groups together.

    $$0+x$$

31. Collect terms with the same power and simplify their coefficients.

    $$x$$

Answer

The final answer is shown below.