Graphs of Piecewise Functions - Free Printable

The image presents a piecewise function and its corresponding graph. Let's analyze and solve the problem step by step.

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🔹 Given Function:

$$
f(x) =
\begin{cases}
2x + 1 & \text{if } x \leq 3 \\
7 & \text{if } x > 3
\end{cases}
$$

This is a piecewise-defined function, meaning it uses different rules depending on the value of $ x $.

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🔹 Step-by-Step Explanation:

#### ✔ Part 1: For $ x \leq 3 $
- The function is defined as $ f(x) = 2x + 1 $
- This is a linear function with slope 2 and y-intercept 1.
- It applies to all $ x $ values less than or equal to 3.
- At $ x = 3 $:
$$
f(3) = 2(3) + 1 = 6 + 1 = 7
$$

#### ✔ Part 2: For $ x > 3 $
- The function is constant: $ f(x) = 7 $
- So, for all $ x $ greater than 3, the output is always 7.
- Note: This part does not include $ x = 3 $ — only values strictly greater than 3.

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🔹 Graph Analysis:

Looking at the graph:

1. Left side (for $ x \leq 3 $):
- A straight line starting from the left, increasing with slope 2.
- Passes through points like:
- $ x = 0 $: $ f(0) = 2(0)+1 = 1 $ → point $ (0,1) $
- $ x = 1 $: $ f(1) = 3 $
- $ x = 2 $: $ f(2) = 5 $
- $ x = 3 $: $ f(3) = 7 $
- The line ends at $ x = 3 $, and since $ x = 3 $ is included in this piece, we use a closed circle at $ (3,7) $

2. Right side (for $ x > 3 $):
- A horizontal line at $ y = 7 $
- Starts just after $ x = 3 $, so we use an open circle at $ (3,7) $ to indicate that $ x = 3 $ is not included in this part.
- But wait! This creates a potential issue...

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⚠️ Important Observation: Continuity at $ x = 3 $

- From the left: $ f(3) = 7 $
- From the right: $ f(x) = 7 $ for $ x > 3 $
- Even though the right-hand piece starts at $ x > 3 $, the value at $ x = 3 $ is already defined in the first piece.

So the function is continuous at $ x = 3 $ because both sides approach 7, and $ f(3) = 7 $. The graph should show:
- A solid dot at $ (3,7) $ (from the left piece)
- A horizontal line extending to the right from $ x = 3 $, but since $ x > 3 $, it starts just after $ x = 3 $

However, in the graph shown:
- There is a solid line going up to $ (3,7) $
- Then a horizontal line continuing to the right from $ (3,7) $
- But no open circle — which suggests the function is continuous there.

✔ This is correct because although the second piece is defined only for $ x > 3 $, the value at $ x = 3 $ is covered by the first piece. So the function is defined and continuous at $ x = 3 $, and the graph can smoothly connect.

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🔹 Final Answer:

The function $ f(x) $ is correctly represented as a piecewise function:

$$
f(x) =
\begin{cases}
2x + 1 & \text{if } x \leq 3 \\
7 & \text{if } x > 3
\end{cases}
$$

And the graph:
- Is a line $ y = 2x + 1 $ for $ x \leq 3 $, ending at $ (3,7) $
- Then a horizontal line $ y = 7 $ for $ x > 3 $, starting just after $ x = 3 $

Since $ f(3) = 7 $, and $ \lim_{x \to 3^+} f(x) = 7 $, the function is continuous at $ x = 3 $.

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✔ Conclusion:

The graph accurately represents the piecewise function. The function has:
- A linear segment from the left up to $ x = 3 $
- A constant value of 7 for $ x > 3 $
- No jump discontinuity at $ x = 3 $

Thus, the function is continuous at $ x = 3 $, and the graph is correct as drawn.

Let me know if you'd like to evaluate $ f(x) $ at specific points or find limits!