The lengths of the sides of a right-angled triangle are given by the expressions $x - 1$, $4x$, and $5x - 9$, as shown in the diagram - Leaving Cert Mathematics - Question 5 - 2016

For $x = 1$:

• Side lengths are:
  • $1 - 1 = 0$ (not valid)

For $x = 10$:

• Side lengths are:
  • $10 - 1 = 9$
  • $4(10) = 40$
  • $5(10) - 9 = 41$

Now we check:

$9^2 + 40^2 = 41^2$

Calculating each term:

• $9^2 = 81$
• $40^2 = 1600$
• $41^2 = 1681$

Thus, $81 + 1600 = 1681$

This confirms that the sides indeed form a Pythagorean triple.

To show that $f(x)$ is injective, we need to demonstrate that: $f(a) = f(b) \Rightarrow a = b$

Let: $f(a) = 3a - 2 \text{ and } f(b) = 3b - 2$

Equating gives: $3a - 2 = 3b - 2$

Solving this: $3a = 3b \Rightarrow a = b$

Thus, $f(x)$ is injective.

To find the inverse function $f^{-1}(x)$:

1. Start with $y = f(x) = 3x - 2$.
2. Swap $x$ and $y$: $x = 3y - 2$
3. Solve for $y$:
   • Add 2 to both sides: $x + 2 = 3y$
   • Divide by 3: $y = \frac{x + 2}{3}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x + 2}{3}$