Mathematics 10 — Exercise 4.1 · Q6 (iv)

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.

   $$y=\frac{x+1}{3\times x-2}$$

2. Multiply both sides by the denominator. The original denominator must remain nonzero.

   $$y\times \left(3\times x-2\right)=x+1$$

3. Expand the left bracket.

   $$3\times x\times y-2\times y=x+1$$

4. Subtract the x term on the right and the y-only term on the left.

   $$3\times x\times y-x=2\times y+1$$

5. Factor out the original input x.

   $$x\times \left(3\times y-1\right)=2\times y+1$$

6. Divide both sides by the bracket, which must be nonzero.

   $$x=\frac{2\times y+1}{3\times y-1}$$

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

   $$\mathrm{f⁻¹}\left(x\right)=\frac{2\times x+1}{3\times x-1}$$

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

   $$(-\infty ,\frac{1}{3})\cup (\frac{1}{3},\infty )$$

Answer

The final answer is shown below.