Which graph represents the piecewise equationsy=5+x𝑦=5+𝑥    (0≤x<5)(0≤𝑥<5)y=2x𝑦=2𝑥    (5≤x≤7)(5≤𝑥≤7)

To analyze the piecewise function given by the equations, we can break down the two equations and their corresponding domains:

1. First Piece:

   • Equation: $y = 5 + x$
   • Domain: $0 \leq x < 5$

2. Second Piece:

   • Equation: $y = 2x$
   • Domain: $5 \leq x \leq 7$

Steps to Graph the Piecewise Function

1. Graph the First Piece:

   • For $0 \leq x < 5$:
     • When $x = 0$: $y = 5 + 0 = 5$
     • When $x = 4.9$: $y = 5 + 4.9 = 9.9$ (approaches but does not include $x = 5$)
     • When $x = 5$: Not included in this piece.
   • Plot the line segment starting from the point (0, 5) to just below the point (5, 10).

2. Graph the Second Piece:

   • For $5 \leq x \leq 7$:
     • When $x = 5$: $y = 2 \times 5 = 10$
     • When $x = 6$: $y = 2 \times 6 = 12$
     • When $x = 7$: $y = 2 \times 7 = 14$
   • Plot the line segment connecting (5, 10) to (7, 14), including both endpoints.

Analysis and Pattern

• The first piece $y = 5 + x$ is a straight line with a positive slope, starting from (0, 5) and increasing.
• The second piece $y = 2x$ also has a positive slope, starting from (5, 10) and continuing to (7, 14).

Verification and Conclusion

• Each piece is continuous in its respective domain. At $x = 5$, the left-hand limit approaches $10$ from the first piece, matching the value for the second piece.
• The lines should be drawn in separate segments to accurately represent the piecewise nature.

In summary, the correct graph will consist of:

• A line segment starting from (0, 5) to just below (5, 10).
• A line segment starting from (5, 10) and including (7, 14).

Final Answer

The graph representing the given piecewise function consists of two segments:

• From $(0, 5)$ to just below $(5, 10)$ for $y = 5 + x$.
• From $(5, 10)$ to $(7, 14)$ for $y = 2x$.