You are given the inequality:

\[
6x - 7 \leq f(x) \leq x^2
\]

This defines bounds for \( f(x) \), meaning that the function \( f(x) \) is squeezed between \( 6x - 7 \) and \( x^2 \) for all values of \( x \).

### Explanation:
- The left side of the inequality, \( 6x - 7 \), provides the lower bound for \( f(x) \).
- The right side of the inequality, \( x^2 \), provides the upper bound for \( f(x) \).
- The function \( f(x) \) must lie between these two expressions for any \( x \) in the domain under consideration.

This inequality can be analyzed by examining specific values of \( x \) or looking at the behavior of \( 6x - 7 \) and \( x^2 \) over different intervals. For example:
- As \( x \to -\infty \), the term \( 6x - 7 \) decreases without bound, while \( x^2 \) increases positively.
- As \( x \to \infty \), \( 6x - 7 \) grows linearly and \( x^2 \) grows quadratically, meaning \( f(x) \) is more tightly constrained at larger values of \( x \).

Do you want more detailed steps on how to find specific values of \( f(x) \) or how to solve it for a particular range of \( x \)?

### Relative Questions:
1. How do \( 6x - 7 \) and \( x^2 \) behave when plotted, and what insights can you gain from the graphs?
2. Can we find specific intervals where \( f(x) \) is strictly within the bounds?
3. How do you solve the inequality if given a specific value of \( f(x) \)?
4. How do quadratic and linear functions differ in growth rate, and how does that affect \( f(x) \)?
5. What happens if the inequality is strict (i.e., \( 6x - 7 < f(x) < x^2 \))?

**Tip**: When working with inequalities involving functions, it's useful to analyze the behavior of both boundary functions (in this case, \( 6x - 7 \) and \( x^2 \)) over different intervals of \( x \) to understand where the inequalities hold.

Explore how to solve the inequality 6x - 7 ≤ f(x) ≤ x^2, which defines bounds for the function f(x) between a linear and a quadratic expression. Learn key mathematical concepts such as algebra, inequalities, and the behavior of linear and quadratic functions. Suitable for students in Grades 9-12.

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