Mathematics 10 — Exercise 4.1 · Q3 (ii) (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.

   $$2\times x^2-1$$

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

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

3. Substitute the function expressions, keeping the brackets.

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

4. Expand this power by multiplying the bracket by itself.

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

5. Distribute the products and combine like terms in this powered bracket.

   $${\left(2\times x^2-1\right)}^2=4\times x^4-4\times x^2+1$$

6. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$8\times x^4-8\times x^2+1$$

7. Write EVERY distributed product before collecting terms.

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

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

   $$-1=-1$$

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

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

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

    $$2\times \left(-1\right)\times \left(2\times x^2\right)=-4\times x^2$$

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

    $$2\times \left(-1\right)\times \left(2\times x^2\right)=-4\times x^2$$

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

    $$2\times \left(2\times x^2\right)\left(2\times x^2\right)=8\times x^4$$

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

    $$-1+2=1$$

14. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 4=-4$$

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

    $$-4-4=-8$$

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

    $$8=8$$

17. Put the simplified groups together.

    $$8\times x^4-8\times x^2+1$$

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

    $$8\times x^4-8\times x^2+1$$

Answer

The final answer is shown below.