Graph each piecewise-defined function. f(x)={−12x2+2if x≤212xif x\u003e2f(x) = \begin{cases} -\frac{1}{2}$$x^2 + 2$$ & \text{if } x \leq 2 \\ \frac{1}{2}x & \text{if } x \u003e 2 \end{cases}

Ch. 2 - Graphs and Functions

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Lial 13th Edition

Ch. 2 - Graphs and Functions

문제 29

3장, 문제 29

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} - \frac{1}{2} x^{2} + 2 & if x \leq 2 \\ \frac{1}{2} x & if x > 2 \end{matrix}$

검증된 단계별 안내

Identify the two pieces of the piecewise function and their respective domains: $f(x) = x^2 - 2$ for $x \leq 1$ and $f(x) = 2x$ for $x > 1$.

Graph the first piece $f(x) = x^2 - 2$ for all $x$ values less than or equal to 1. This is a parabola shifted down by 2 units. Plot points including $x=1$ to find the boundary value.

Graph the second piece $f(x) = 2x$ for all $x$ values greater than 1. This is a straight line with slope 2. Start plotting from just greater than $x=1$ to the right.

At $x=1$, evaluate both pieces to check if the function is continuous. Calculate $f(1)$ from the first piece and the limit from the second piece as $x$ approaches 1 from the right.

Use open or closed circles to indicate whether the point at $x=1$ is included in the graph for each piece: closed circle for $x \leq 1$ and open circle for $x > 1$. Then connect the points smoothly according to the function definitions.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Piecewise-Defined Functions

A piecewise-defined function is a function composed of different expressions depending on the input value's domain. Each piece applies to a specific interval, and the function's overall graph combines these pieces. Understanding how to interpret and graph each piece separately is essential.

Graphing Quadratic and Linear Functions

The function includes a quadratic expression (x² - 2) for x ≤ 1 and a linear expression (2x) for x > 1. Knowing how to graph quadratic functions (parabolas) and linear functions (straight lines) helps in plotting each piece accurately on the coordinate plane.

Domain Restrictions and Continuity

Each piece of the function is defined over a specific domain (x ≤ 1 or x > 1). Understanding domain restrictions ensures correct graphing boundaries. Additionally, checking the function's value at the boundary (x = 1) helps determine if the graph is continuous or has a jump.

Graph each piecewise-defined function. f(x)={−12x2+2if x≤212xif x>2f(x) =\(\begin{cases}\)-\(\frac{1}{2}\)x^2 + 2 & \(\text{if }\) x \(\leq\) 2 \(\frac{1}{2}\)x & \(\text{if }\) x > 2\(\end{cases}\)