The question asks to find the value of $f(-7)$ for the function $f(x) = -x^2$ and explain the reasoning.

Solution:

1. Substitute $x = -7$ into the function $f(x) = -x^2$: $f(-7) = -(-7)^2$

2. Simplify the expression: $(-7)^2 = 49$ So: $f(-7) = -49$

Final Answer:

The value of $f(-7)$ is $-49$.

Explanation:

The function $f(x) = -x^2$ means that for any value of $x$, the square of $x$ is first computed and then multiplied by $-1$. Substituting $x = -7$, we compute $(-7)^2 = 49$, then multiply it by $-1$, resulting in $-49$.

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Here are 5 related questions to deepen understanding:

1. What is the value of $f(5)$ for $f(x) = -x^2$?
2. Why is the result always negative for this function $f(x) = -x^2$?
3. How does the graph of $f(x) = -x^2$ compare to $g(x) = x^2$?
4. What is the domain and range of $f(x) = -x^2$?
5. How would the function $f(x) = -x^2$ change if written as $f(x) = -(x+1)^2$?

Tip: Always follow the order of operations carefully, especially when dealing with powers and negatives.