Question

Select the correct answer.
What is the range of g(x) = 3|x − 1| − 1?

A.
(-∞, 1]
B.
[-1, ∞)
C.
[1, ∞)
D.
(-∞, ∞)

Answer 1

To find the range of the function \( g(x) = 3|x - 1| - 1 \), we first analyze the expression \( 3|x - 1| \).

1. The absolute value \( |x - 1| \) is always non-negative, meaning \( |x - 1| \geq 0 \).
2. Therefore, \( 3|x - 1| \geq 0 \).
3. This implies that \( 3|x - 1| - 1 \) will reach its minimum value when \( |x - 1| = 0 \).

Let's find this minimum value:

- The minimum value of \( |x - 1| \) is 0 (which occurs when \( x = 1 \)).
- Hence, at \( x = 1 \):
\[
g(1) = 3(0) - 1 = -1
\]

Now, as \( |x - 1| \) increases (as \( x \) moves away from 1), \( g(x) \) will also increase. As \( |x - 1| \) can become infinitely large, \( g(x) \) will also tend to infinity.

Thus, we see that:

- The minimum value of \( g(x) \) is \( -1 \).
- There is no upper limit; \( g(x) \) can grow arbitrarily large.

Combining these observations, the range of \( g(x) \) is:

\[
[-1, \infty)
\]

Thus, the correct answer is **B. \([-1, \infty)\)**.