To determine the correct answer for this question, let’s analyze each function and option.

1. Given Functions:

   • The linear function $f(x) = x + 1$: This is a straight line with a slope of 1 and y-intercept of 1.
   • The absolute value function $f(x) = |x| - 1$: This creates a V-shaped graph, with its vertex at $(0, -1)$.

2. Domain and Range of Each Function:

   • Domain:
     • For both $f(x) = x + 1$ and $f(x) = |x| - 1$, the domain is all real numbers, $(-\infty, \infty)$, because both functions accept any real number as input.
   • Range:
     • For $f(x) = x + 1$, the range is also all real numbers $(-\infty, \infty)$, as a linear function has no vertical restrictions.
     • For $f(x) = |x| - 1$, the range is $[-1, \infty)$, since the absolute value function ensures that $f(x) \geq -1$ for all $x$.

3. Analysis of Answer Choices:

   • Option A: "The range of both functions is $(-\infty, \infty)$."
     • This is incorrect because the range of $f(x) = |x| - 1$ does not include values below $-1$.
   • Option B: "The domain of both functions is $(-\infty, \infty)$."
     • This is correct, as both functions can take any real number as input.
   • Option C: "The domain of both functions is $[1, \infty)$."
     • This is incorrect because the domain of both functions is not restricted to positive values; they accept all real numbers.
   • Option D: "The range of both functions is $[1, \infty)$."
     • This is incorrect because the range of $f(x) = |x| - 1$ starts from $-1$, not 1.

4. Conclusion:

   • The correct answer is Option B, "The domain of both functions is $(-\infty, \infty)$".

Would you like further details or have any questions?

Follow-Up Questions:

1. What is the effect of changing the slope in a linear function?
2. How would the range of $f(x) = |x| - 1$ change if the -1 were replaced with another constant?
3. What would be the range of $f(x) = |x + 1|$?
4. How do transformations affect the vertex of an absolute value function?
5. How does the graph of $f(x) = |x|$ differ from $f(x) = |x| - 1$?

Tip: When working with absolute value functions, pay attention to the vertex and how transformations (like shifts) impact it.