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

   $$\frac{2}{x}$$

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

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

3. Substitute the function expressions, keeping the brackets.

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

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

   $$-1+\frac{8}{x^2}$$

5. Write EVERY distributed product before collecting terms.

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

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

   $$-1=-1$$

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

   $$2\times {\left(\frac{2}{x}\right)}^2=\frac{8}{x^2}$$

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

   $$-1=-1$$

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

   $$8=8$$

10. Put the simplified groups together.

    $$-1+\frac{8}{x^2}$$

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

    $$-1+\frac{8}{x^2}$$

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

    $$(x\ne 0,x^2\ne 0)$$

Answer

The final answer is shown below.