Mathematics 10 — Exercise 4.1 · Q6 (iii)

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.

   $$y=\sqrt{x-1}$$

2. A square-root output is nonnegative: y ≥ 0.

   $$y\ge 0$$

3. Square both sides to remove the square root.

   $$y^2=x-1$$

4. Add the shift to both sides.

   $$x=y^2+1$$

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

   $$\mathrm{f⁻¹}\left(x\right)=x^2+1$$

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

   $$[0,\infty )$$

Answer

The final answer is shown below.