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

Full question

For the following functions, find (fof)(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.

   $$2\times \left(2\times x-1\right)-1$$

3. Substitute the function expressions, keeping the brackets.

   $$2\times \left(2\times x-1\right)-1$$

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

   $$4\times x-3$$

5. Write EVERY distributed product before collecting terms.

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

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

   $$-1=-1$$

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

   $$2\times \left(-1\right)=-2$$

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

   $$2\times 2\times x=4\times x$$

9. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times 2=-2$$

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

    $$-1-2=-3$$

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

    $$4=4$$

12. Put the simplified groups together.

    $$4\times x-3$$

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

    $$4\times x-3$$

Answer

The final answer is shown below.