The function given is:

$y = |x| + 1$

Analysis of the Function:

1. Domain:
   The absolute value function $|x|$ is defined for all real numbers, so the domain is:
   $(-\infty, \infty)$

2. Range:
   Since the absolute value function $|x|$ is always non-negative ($|x| \geq 0$), adding 1 shifts the graph up by 1 unit. Thus, the smallest possible value of $y$ is 1, giving the range:
   $[1, \infty)$

3. Graph Shape:

   • The function is V-shaped because of the absolute value.
   • The vertex (lowest point) is at $(0,1)$.
   • For $x > 0$, the function behaves as $y = x + 1$ (a straight line with slope 1).
   • For $x < 0$, the function behaves as $y = -x + 1$ (a straight line with slope -1).

4. Intercepts:

   • Y-intercept: Set $x = 0$, then $y = 1$, so the y-intercept is (0,1).
   • X-intercepts: Solve $|x| + 1 = 0$, which has no solution, meaning there are no x-intercepts.

Graphing Steps:

1. Plot the vertex at $(0,1)$.
2. Draw two linear pieces:
   • A line with slope $+1$ for $x > 0$.
   • A line with slope $-1$ for $x < 0$.
3. The result is a symmetric "V" shape with the vertex at $(0,1)$.

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Related Questions:

1. How do transformations affect the absolute value function?
2. What is the effect of adding different constants to $|x|$?
3. How do we find the inverse of $y = |x| + 1$?
4. What happens if we modify it to $y = a|x| + b$?
5. How do we express $|x|$ as a piecewise function?

Tip: The function $y = |x| + k$ always shifts the basic absolute value graph up or down by $k$, affecting the range but not the domain.