Mathematics 10 — Exercise 4.1 · Q4

Full question

Find k so (fog)(x)=(gof)(x), with f(x)=3x+2 and g(x)=6x−k.

Step-by-step solution

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

   $$-k+6\times x$$

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

   $$3\times \left(-k+6\times x\right)+2$$

3. Substitute the function expressions, keeping the brackets.

   $$3\times \left(-k+6\times x\right)+2$$

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

   $$-3\times k+18\times x+2$$

5. Write EVERY distributed product before collecting terms.

   $$2+3\times \left(-1\right)\times k+3\times 6\times x$$

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

   $$2=2$$

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

   $$3\times \left(-1\right)\times k=-3\times k$$

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

   $$3\times 6\times x=18\times x$$

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

   $$2=2$$

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

    $$-3=-3$$

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

    $$18=18$$

12. Put the simplified groups together.

    $$-3\times k+18\times x+2$$

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

    $$-3\times k+18\times x+2$$

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

    $$3\times x+2$$

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

    $$-k+6\times \left(3\times x+2\right)$$

16. Substitute the function expressions, keeping the brackets.

    $$-k+6\times \left(3\times x+2\right)$$

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

    $$-k+18\times x+12$$

18. Write EVERY distributed product before collecting terms.

    $$-k+6\times 2+6\times 3\times x$$

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

    $$-k=-k$$

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

    $$6\times 2=12$$

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

    $$6\times 3\times x=18\times x$$

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

    $$-1=-1$$

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

    $$12=12$$

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

    $$18=18$$

25. Put the simplified groups together.

    $$-k+12+18\times x$$

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

    $$-k+18\times x+12$$

27. Equate the two compositions for every x.

    $$-3\times k+18\times x+2=-k+18\times x+12$$

28. Equate the corresponding entries.

    $$-3\times k+18\times x+2=-k+18\times x+12$$

29. Collect the variable terms on the left and constants on the right.

    $$-2\times k=18\times x+12-(18\times x+2)$$

30. Simplify the constant calculation.

    $$-2\times k=10$$

31. Divide both sides by the variable coefficient.

    $$k=\frac{10}{-2}$$

32. Reduce the fraction.

    $$k=-5$$

Answer

The final answer is shown below.