The three roots of the equation f(x) = 0 are x = {-2, 0, 3}. What are the three values of x for which f (x - 3) = 0?

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.