Mathematics 10 — Exercise 4.1 · Q6 (ii)

Full question

Find the inverse function f⁻¹(x) of the following function f(x), stating any restrictions on its domain.

Step-by-step solution

1. Write the output as y. To find the inverse, solve this equation for the original input x.

   $$y=4\times x^3-1$$

2. Subtract the constant from both sides.

   $$y+1=4\times x^3$$

3. Divide both sides by the input coefficient.

   $$x^3=\frac{y+1}{4}$$

4. Take the real cube root on both sides. A cube has exactly one real cube root.

   $$x=\sqrt[3]{\frac{y}{4}+\frac{1}{4}}$$

5. Interchange the names x and y: the old output is now the inverse input.

   $$\mathrm{f⁻¹}\left(x\right)=\sqrt[3]{\frac{x}{4}+\frac{1}{4}}$$

6. The inverse domain is the original range. Keep any exclusions or nonnegative-input restriction.

   $$\mathrm{ℝ}$$

Answer

The final answer is shown below.