Question

Graph each piecewise-defined function. f(x)={x−1if x≤32if x\u003e3f(x) =
\begin{cases}
x - 1 & 	ext{if } x \leq 3 \
2 & 	ext{if } x \u003e 3
\end{cases}

Accepted Answer

Step 1: Understand the piecewise function definition. The function f(x) is defined as two different expressions depending on the value of x: for x ≤ 1, f(x) = x + 4; for x \u003e 1, f(x) = 5. Step 2: Graph the first piece, f(x) = x + 4 for x ≤ 1. This is a linear function with slope 1 and y-intercept 4. Plot points for values of x less than or equal to 1, such as x = 0, x = 1, and connect them with a straight line extending to the left. Step 3: Determine the point at x = 1 for the first piece. Calculate f(1) = 1 + 4 = 5. This point (1, 5) will be included in the graph of the first piece because the inequality is 'less than or equal to'. Step 4: Graph the second piece, f(x) = 5 for x \u003e 1. This is a constant function, so draw a horizontal line at y = 5 for all x values greater than 1. Use an open circle at (1, 5) if you want to indicate that this piece does not include x = 1, but since the first piece includes x = 1, the point at (1, 5) belongs to the first piece. Step 5: Combine both pieces on the same coordinate plane. The graph will show a line with slope 1 up to and including x = 1, and a horizontal line at y = 5 for x values greater than 1. This completes the graph of the piecewise function.

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} x - 1 & if x \leq 3 \\ 2 & if x > 3 \end{matrix}$

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Key Concepts

Piecewise-Defined Functions

Graphing Functions with Domain Restrictions

Continuity and Discontinuity in Piecewise Functions

Graph each piecewise-defined function. f(x)={x−1if x≤32if x>3f(x) =\(\begin{cases}\)x - 1 & \(\text{if }\) x \(\leq\) 3 \\2 & \(\text{if }\) x > 3\(\end{cases}\)