Piecewise Function Equation from Graph

Based on the image provided, the problem asks for the algebraic equation of a piecewise function $f(x)$ shown on a Cartesian plane.

The graph consists of three distinct segments:

1. A parabolic curve on the left with a vertex at the origin, ending at $x = 1$.
2. A horizontal line segment in the middle between $x = 1$ and $x = 2$.
3. A linear ray starting from $x = 2$ and extending to the right.

Answer

The piecewise function is composed of a quadratic part for $x \le 1$, a constant part for $1 < x \le 2$, and a linear part for $x > 2$. The complete analytical expression is:

$f(x) = \begin{cases} x^2 & \text{if } x \le 1 \\ 3 & \text{if } 1 < x \le 2 \\ x & \text{if } x > 2 \end{cases}$

Explanation

1. Identify the quadratic segment ($x \le 1$) The leftmost part of the graph is a parabola that passes through $(0,0)$, $(-1,1)$, $(-2,4)$, and $(1,1)$. Since the vertex is at $(0,0)$, we use the parent form $y = ax^2$. Substituting $(1,1)$ gives $1 = a(1)^2$, so $a = 1$. $y = x^2$ This formula represents a standard parabola with its turning point at the origin. ⚠️ This step is required on exams: Observe the solid dot at $(1,1)$, which indicates the interval is inclusive ($x \le 1$).

2. Identify the constant segment ($1 < x \le 2$) The middle portion is a horizontal line at the height of $y = 3$. It starts with an open circle at $x = 1$ (exclusive) and ends with a solid dot at $x = 2$ (inclusive). $y = 3$ This indicates that for any input value between 1 and 2, the output is always exactly 3.

3. Identify the linear segment ($x > 2$) The final part starts with an open circle at $(2,2)$ and passes through $(3,3)$ and $(4,4)$. We calculate the slope $m$ using $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{3 - 2} = 1$. Using the point-slope form $y - 2 = 1(x - 2)$, we simplify to $y = x$. $y = x$ This formula represents an identity line where the output value is equal to the input value. ⚠️ This step is required on exams: Use the open circle at $(2,2)$ to define the domain as $x > 2$.

Final Answer

The equation of the piecewise function is: $\boxed{f(x) = \begin{cases} x^2 & x \le 1 \\ 3 & 1 < x \le 2 \\ x & x > 2 \end{cases}}$

Common Mistakes

• Incorrect Inequality Symbols: Students often confuse solid dots ($\le$ or $\ge$) with open circles ($<$ or $>$). Always verify which endpoint is "filled in."
• Domain Overlap: A relation is not a function if the domains overlap (e.g., using $x \le 1$ and $1 \le x \le 2$). Each value of $x$ must belong to exactly one piece.