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

Full question

For the following functions, find (gof)(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.

   $$2\times x-1$$

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

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

3. Substitute the function expressions, keeping the brackets.

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

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

   $$x$$

5. Write EVERY distributed product before collecting terms.

   $$\frac{1}{2}+\frac{1}{2}\times \left(-1\right)+\frac{2\times x}{2}$$

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.

    $$\frac{1\times \left(-1\right)}{2\times 1}$$

11. Calculate these two products separately.

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

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

    $$\frac{1}{2}\times \left(-1\right)=-\frac{1}{2}$$

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

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

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

    $$\frac{\left(-1\right)\times 1}{1\times 2}$$

15. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{2}=-\frac{1}{2}$$

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

    $$\mathrm{LCD}=2$$

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

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

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

    $$-\frac{1}{2}=\frac{\left(-1\right)\times 1}{2\times 1}$$

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

    $$\frac{1-1}{2}=\frac{0}{2}$$

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

    $$\mathrm{GCD}=2$$

21. Show the two divisions explicitly.

    $$\frac{\frac{0}{2}}{\frac{2}{2}}$$

22. Evaluate the numerator division.

    $$\frac{0}{2}=0$$

23. Evaluate the denominator division.

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

24. Write the reduced fraction.

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

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

    $$\frac{1}{2}-\frac{1}{2}=0$$

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

    $$1=1$$

27. Put the simplified groups together.

    $$0+x$$

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

    $$x$$

Answer

The final answer is shown below.