Graph each piecewise-defined function. f(x)={4−xif x\u003c21+2xif x≥2f(x) = \begin{cases} 4 - x & \text{if } x \u003c 2 \\ 1 + 2x & \text{if } x \geq 2 \end{cases}

Ch. 2 - Graphs and Functions

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Lial 13th Edition

Ch. 2 - Graphs and Functions

문제 23

3장, 문제 23

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} 4 - x & if x < 2 \\ 1 + 2 x & if x \geq 2 \end{matrix}$

검증된 단계별 안내

Identify the two pieces of the piecewise function: $f(x) = x + 5$ for $x \leq 3$ and $f(x) = 14 - 3x$ for $x > 3$.

Graph the first piece $f(x) = x + 5$ for all $x$ values less than or equal to 3. This is a line with slope 1 and y-intercept 5. Plot points such as $(0, 5)$ and $(3, 8)$ and draw the line up to $x = 3$.

At $x = 3$, mark the point on the first piece as a closed dot since the inequality includes $x = 3$. The point is $(3, 8)$ because $3 + 5 = 8$.

Graph the second piece $f(x) = 14 - 3x$ for $x > 3$. This is a line with slope $-3$ and y-intercept 14. Plot points such as $(4, 2)$ and $(5, -1)$ and draw the line starting just to the right of $x = 3$.

At $x = 3$, mark the point on the second piece as an open dot since the inequality is strict ($x > 3$). Calculate $f(3)$ for the second piece to find the open dot location: $14 - 3(3) = 5$.

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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 behavior is determined by these separate rules.

Graphing Piecewise Functions

To graph a piecewise function, plot each piece on its respective domain interval. Pay attention to endpoints, using open or closed circles to indicate whether points are included or excluded, ensuring continuity or jumps are accurately represented.

Domain and Interval Notation

Understanding domain restrictions is crucial for piecewise functions. Each piece is valid only over a specified interval, often expressed using inequalities. Correctly interpreting these intervals ensures the function is graphed and analyzed properly.

Graph each piecewise-defined function. f(x)={4−xif x<21+2xif x≥2f(x) = \(\begin{cases}\) 4 - x & \(\text{if }\) x < 2 \\ 1 + 2x & \(\text{if }\) x \(\geq\) 2 \(\end{cases}\)