Mathematics 10 — Exercise 4.1 · Q3 (iii) (d)

Full question

For the following functions, find (gog)(x).

Step-by-step solution

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

   $$\frac{x}{2}+\frac{1}{2}$$

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

   $$\frac{\frac{x}{2}+\frac{1}{2}}{2}+\frac{1}{2}$$

3. Substitute the function expressions, keeping the brackets.

   $$\frac{\frac{x}{2}+\frac{1}{2}}{2}+\frac{1}{2}$$

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

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

5. Write EVERY distributed product before collecting terms.

   $$\frac{1}{2}+\frac{1}{2}\times \frac{1}{2}+\frac{\frac{x}{2}}{2}$$

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

   $$\frac{1}{2}=\frac{1}{2}$$

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

   $$\frac{1\times 1}{1\times 2}$$

8. Calculate these two products separately.

   $$\frac{1}{2}$$

9. Calculate this operation before combining the remaining terms.

   $$1\times \frac{1}{2}=\frac{1}{2}$$

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

    $$\frac{1\times 1}{2\times 2}$$

11. Calculate these two products separately.

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

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

    $$\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$$

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

    $$\frac{\frac{x}{2}}{2}=\frac{x}{4}$$

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

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

15. Calculate this operation before combining the remaining terms.

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

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

    $$\mathrm{LCD}=4$$

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

    $$\frac{1}{2}=\frac{1\times 2}{2\times 2}$$

18. Calculate the converted numerator; the denominator is now common.

    $$\frac{2}{4}$$

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

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

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

    $$\frac{2+1}{4}=\frac{3}{4}$$

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

    $$\frac{1}{2}+\frac{1}{4}=\frac{3}{4}$$

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

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

23. Put the simplified groups together.

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

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

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

25. Retain inputs where the inner function is defined and where its output makes the outer function defined. Neither denominator may be zero.

    $$(1\ne 0,4\ne 0)$$

Answer

The final answer is shown below.