1. The problem is to summarize the main points of a function concept typically denoted by $f(x)$. 2. A function $f(x)$ is a rule that assigns to each element $x$ in the domain exactly one element $f(x)$ in the codomain. 3. Main points include: - **Domain**: The set of all possible input values $x$. - **Range**: The set of all possible output values $f(x)$. - **Intercepts**: Points where the function crosses the axes. The $y$-intercept is $f(0)$ and $x$-intercepts are solutions to $f(x)=0$. - **Increasing/Decreasing**: Intervals where $f(x)$ rises or falls as $x$ increases. - **Extrema**: Local maxima and minima where the function reaches peaks or valleys. - **Continuity**: Whether $f(x)$ is unbroken over its domain. - **Behavior at infinity**: Limits of $f(x)$ as $x \to \pm\infty$. 4. To illustrate, the function $f(x) = x^2$ has: - Domain: All real numbers $\mathbb{R}$ - Range: $[0, \infty)$ - $y$-intercept: $(0,0)$ - $x$-intercept: $(0,0)$ - Increasing on $(0, \infty)$ and decreasing on $(-\infty, 0)$ - Minimum at $(0,0)$ - Continuous everywhere - Tends to $\infty$ as $x \to \pm\infty$ 5. Understanding these aspects helps in analyzing and sketching the graph of any function $f(x)$. Final answer: The key concept of a function $f(x)$ involves its domain, range, intercepts, increasing/decreasing intervals, extrema, continuity, and end behavior, crucial for graph interpretation and analysis.