Graph each piecewise-defined function.f(x)={2x+1if x≥0xif x\u003c0f(x) = \begin{cases} 2x + 1 & \text{if } x \geq 0 \\ x & \text{if } x \u003c 0 \end{cases}

Ch. 2 - Graphs and Functions

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Lial 13th Edition

Ch. 2 - Graphs and Functions

문제 24

3장, 문제 24

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} 2 x + 1 & if x \geq 0 \\ x & if x < 0 \end{matrix}$

검증된 단계별 안내

Identify the two pieces of the piecewise function: $f(x) = 4x - 2$ for $x \leq 0$ and $f(x) = 2x$ for $x > 0$.

For the first piece $f(x) = 4x - 2$ when $x \leq 0$, find key points by substituting values of $x$ less than or equal to 0, such as $x = 0$ and $x = -1$.

Plot the points from the first piece on the coordinate plane and draw a line through them, making sure to include the point at $x=0$ with a solid dot since the inequality includes equality ($\leq$).

For the second piece $f(x) = 2x$ when $x > 0$, find points by substituting values of $x$ greater than 0, such as $x = 1$ and $x = 2$.

Plot these points and draw a line through them, using an open circle at $x=0$ to indicate that this point is not included in the second piece since the inequality is strict ($>$).

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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 is formed by combining these pieces. Understanding how to interpret and graph each piece separately is essential.

Graphing Linear Functions

Each piece of the function is linear, represented by an equation of the form y = mx + b. To graph these, identify the slope (m) and y-intercept (b), then plot points accordingly. For piecewise functions, graph each linear piece only over its specified domain.

Domain Restrictions and Continuity

The domain restrictions (e.g., x ≤ 0 or x > 0) determine where each piece of the function applies. When graphing, ensure to respect these boundaries, using closed or open dots to indicate inclusion or exclusion of endpoints. This helps in understanding the function's continuity and behavior at boundary points.

Graph each piecewise-defined function.f(x)={2x+1if x≥0xif x<0f(x) = \(\begin{cases}\) 2x + 1 & \(\text{if }\) x \(\geq\) 0 \\ x & \(\text{if }\) x < 0 \(\end{cases}\)