Mathematics 10 — Exercise 4.1 · Q8 (iii)

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.

   $$y=\frac{x}{4}-\frac{3}{4}$$

2. Subtract the constant from both sides.

   $$y+\frac{3}{4}=\frac{x}{4}$$

3. Divide both sides by the input coefficient.

   $$x=\frac{1}{\frac{1}{4}}\times \left(y+\frac{3}{4}\right)$$

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

   $$\mathrm{f⁻¹}\left(x\right)=4\times x+3$$

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.

   $$4\times x+3$$

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

   $$\frac{4\times x+3}{4}-\frac{3}{4}$$

9. Substitute the function expressions, keeping the brackets.

   $$\frac{4\times x+3}{4}-\frac{3}{4}$$

10. Distribute each bracket product: each term multiplies every term in the other bracket.

    $$x$$

11. Write EVERY distributed product before collecting terms.

    $$-\frac{3}{4}+\frac{1}{4}\times 3+\frac{4\times x}{4}$$

12. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-3\right)\times 1}{1\times 4}$$

13. Calculate these two products separately.

    $$-\frac{3}{4}$$

14. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \frac{1}{4}=-\frac{3}{4}$$

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

    $$-\frac{3}{4}=-\frac{3}{4}$$

16. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 4}$$

17. Calculate these two products separately.

    $$\frac{1}{4}$$

18. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{4}=\frac{1}{4}$$

19. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 3}{4\times 1}$$

20. Calculate these two products separately.

    $$\frac{3}{4}$$

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

    $$\frac{1}{4}\times 3=\frac{3}{4}$$

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

    $$\frac{4\times x}{4}=x$$

23. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{3\times 1}{1\times 4}$$

24. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{4}=\frac{3}{4}$$

25. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=4$$

26. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{3}{4}=\frac{\left(-3\right)\times 1}{4\times 1}$$

27. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{3}{4}=\frac{3\times 1}{4\times 1}$$

28. Keep the common denominator and add the signed numerators.

    $$\frac{-3+3}{4}=\frac{0}{4}$$

29. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=4$$

30. Show the two divisions explicitly.

    $$\frac{\frac{0}{4}}{\frac{4}{4}}$$

31. Evaluate the numerator division.

    $$\frac{0}{4}=0$$

32. Evaluate the denominator division.

    $$\frac{4}{4}=1$$

33. Write the reduced fraction.

    $$0\times \frac{1}{1}$$

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

    $$-\frac{3}{4}+\frac{3}{4}=0$$

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

    $$1=1$$

36. Put the simplified groups together.

    $$0+x$$

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

    $$x$$

38. Compute f⁻¹(f(x)) independently.

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

    $$\frac{x}{4}-\frac{3}{4}$$

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

    $$4\times \left(\frac{x}{4}-\frac{3}{4}\right)+3$$

41. Substitute the function expressions, keeping the brackets.

    $$4\times \left(\frac{x}{4}-\frac{3}{4}\right)+3$$

42. Distribute each bracket product: each term multiplies every term in the other bracket.

    $$x$$

43. Write EVERY distributed product before collecting terms.

    $$3+4\times \left(-\frac{3}{4}\right)+4\times \left(\frac{1}{4}\right)\times x$$

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

    $$3=3$$

45. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{4\times \left(-3\right)}{1\times 4}$$

46. Calculate these two products separately.

    $$-\frac{12}{4}$$

47. Show the two divisions explicitly.

    $$\frac{-\frac{12}{4}}{\frac{4}{4}}$$

48. Evaluate the numerator division.

    $$-\frac{12}{4}=-3$$

49. Write the reduced fraction.

    $$-3\times \frac{1}{1}$$

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

    $$4\times \left(-\frac{3}{4}\right)=-3$$

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

    $$4\times \left(\frac{1}{4}\right)\times x=x$$

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

    $$3-3=0$$

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

    $$1=1$$

54. Put the simplified groups together.

    $$0+x$$

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

    $$x$$

Answer

The final answer is shown below.