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

Full question

For the following functions, find (fog)(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.

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

3. Substitute the function expressions, keeping the brackets.

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

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

   $$x$$

5. Write EVERY distributed product before collecting terms.

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

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

   $$-1=-1$$

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

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

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

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

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

   $$-1+1=0$$

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

    $$1=1$$

11. Put the simplified groups together.

    $$0+x$$

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

    $$x$$

Answer

The final answer is shown below.