Understanding Function Transformation and Roots

The question asks us to find the new roots of a transformed function, specifically $f(x - 3) = 0$, given the roots of the original function $f(x) = 0$. The original roots are provided as $x = \{-2, 0, 3\}$.

What is a Root?

A root of an equation $f(x) = 0$ is a value of $x$ that makes the equation true, meaning $f(x)$ evaluates to zero. For the given equation $f(x) = 0$, the roots are the specific $x$-values where the function's graph intersects the x-axis.

Understanding the Transformation $f(x - 3)$

The transformation from $f(x)$ to $f(x - 3)$ represents a horizontal shift of the function's graph. Specifically, replacing $x$ with $(x - 3)$ shifts the entire graph 3 units to the right.

Think about it this way: If $f(a) = 0$ (meaning 'a' is a root of $f(x)$), then for $f(x - 3)$ to be zero, the input to the function, which is $(x - 3)$, must be equal to 'a'. So, we set the new input $(x - 3)$ equal to the old roots.

Calculating the New Roots

We need to find the values of $x$ for which $f(x - 3) = 0$. This means the expression $(x - 3)$ must take on the values of the original roots.

• Original root 1: $x = -2$
• Original root 2: $x = 0$
• Original root 3: $x = 3$

We set the argument of the new function, $(x - 3)$, equal to each of the original roots:

1. Step 1: Set $x - 3$ equal to the first original root (-2).

   $$x - 3 = -2$$

   Add 3 to both sides to solve for $x$:

   $$x = -2 + 3$$

   $$x = 1$$

2. Step 2: Set $x - 3$ equal to the second original root (0).

   $$x - 3 = 0$$

   Add 3 to both sides to solve for $x$:

   $$x = 0 + 3$$

   $$x = 3$$

3. Step 3: Set $x - 3$ equal to the third original root (3).

   $$x - 3 = 3$$

   Add 3 to both sides to solve for $x$:

   $$x = 3 + 3$$

   $$x = 6$$

Conclusion

By applying the horizontal shift transformation, we find that the new values of $x$ for which $f(x - 3) = 0$ are $1$, $3$, and $6$. This corresponds to shifting the original roots $\{-2, 0, 3\}$ three units to the right on the number line.