From the uploaded image, we are solving the following problem:

1. Given the table values for a linear function $f(x)$:

   \hline x & f(x) \\ \hline -4 & 3.5 \\ 0 & 1.5 \\ 1 & 1 \\ 5 & -1 \\ \hline \end{array}$$ Find which functions have the same $$x$$-intercept as $$f(x)$$.

Step 1: Find the $x$-intercept of $f(x)$

To find the $x$-intercept of $f(x)$, set $f(x) = 0$:

1. From the table, observe that $f(x)$ decreases linearly.
2. Between $x = 1$ and $x = 5$, $f(x)$ transitions from $1$ to $-1$.
3. The $x$-intercept occurs when $f(x) = 0$, i.e., between $x = 1$ and $x = 5$.
4. Using interpolation: $x_{\text{intercept}} = 1 + \frac{1}{1 - (-1)} = 1.5$

Thus, $x_{\text{intercept}} = 1.5$.

Step 2: Check each function's $x$-intercept

• For $g(x) = -2x + 5$: Set $g(x) = 0$, $0 = -2x + 5 \implies x = \frac{5}{2} = 2.5.$ Not the same as $f(x)$.

• For $h(x) = (x - 3)^2$: Set $h(x) = 0$, $0 = (x - 3)^2 \implies x = 3.$ Not the same as $f(x)$.

• For $k(x) = (x + 3)^2 + 1$: Set $k(x) = 0$, $0 = (x + 3)^2 + 1 \implies \text{No real solutions}.$ Not the same as $f(x)$.

• For $m(x) = 2x - 6$: Set $m(x) = 0$, $0 = 2x - 6 \implies x = 3.$ Not the same as $f(x)$.

Conclusion

None of the provided functions have the same $x$-intercept as $f(x)$.

Let me know if you'd like further clarification or detailed steps!

Follow-up questions for exploration:

1. How do we find the slope of $f(x)$ from the given table?
2. What is the general form of a linear function?
3. How do quadratic functions differ in their $x$-intercepts compared to linear ones?
4. What methods exist to solve for $x$-intercepts of nonlinear equations?
5. How does interpolation work to estimate values between data points?

Tip: Always cross-check calculations for $x$-intercepts by plotting functions when possible!