Question

What are the domain and range of this function?
y=|3x–3|–5

Answer 1

To find the domain and range of the function \( y = |3x - 3| - 5 \), we will analyze both components separately.

### Domain
The expression \( |3x - 3| \) is defined for all real numbers since absolute value is defined for every real input. Therefore, the domain of the function is:

\[
\text{Domain} = (-\infty, \infty)
\]

### Range
Next, we will analyze the range of the function. The function can be broken down as follows:

1. The expression inside the absolute value, \( 3x - 3 \), can take any real value. The absolute value \( |3x - 3| \) will therefore be non-negative, meaning:

\[
|3x - 3| \geq 0
\]

2. Consequently, the minimum value of \( |3x - 3| \) is \( 0 \), which occurs when \( 3x - 3 = 0 \) or \( x = 1 \).

3. Since \( |3x - 3| \) can be \( 0 \) or greater, \( |3x - 3| - 5 \) gives:

\[
|3x - 3| - 5 \geq 0 - 5 = -5
\]

Thus, the function \( y = |3x - 3| - 5 \) can attain a minimum value of \( -5 \).

4. As \( |3x - 3| \) increases without bound, \( y \) can also increase without bound. Therefore, the function can reach values infinitely greater than \( -5 \).

Combining these observations, we find that the range of the function is:

\[
\text{Range} = [-5, \infty)
\]

### Summary
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [-5, \infty) \)