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.