Question

What are the domain and range of this function?
y=|3x–12|–7

Answer 1

To determine the domain and range of the function \( y = |3x - 12| - 7 \), let's analyze each part.

### Domain:
The expression \( 3x - 12 \) is a linear expression, and the absolute value function is defined for all real numbers. Therefore, the domain of the function is all real numbers.

**Domain:** \( (-\infty, \infty) \)

### Range:
To find the range, we first need to analyze the absolute value part of the function.

1. The expression \( |3x - 12| \) gives us the distance from \( 3x - 12 \) to zero. This expression is always non-negative, meaning it can take values from 0 to \( \infty \).

2. So, the minimum value of \( |3x - 12| \) is 0, which occurs when \( 3x - 12 = 0 \) or \( x = 4 \). At this \( x \)-value:
\[
y = |3(4) - 12| - 7 = |0| - 7 = -7.
\]

3. As \( |3x - 12| \) increases from 0, \( y \) will increase from \(-7\). There is no upper limit to the values of \( |3x - 12| \), thus \( y \) can go to infinity as \( |3x - 12| \) approaches infinity.

This means:
\[
y = |3x - 12| - 7 \geq -7.
\]

Consequently, the range is all values \( y \) can take starting from \(-7\) and going up to infinity.

**Range:** \( [-7, \infty) \)

In summary:
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [-7, \infty) \)