Questions: What are the domain and range of this function? y=4x+1 domain: x x<0, range: y y<-1 domain: x x<0, range: y y ≥ 0 domain: all real numbers, range: y y<-1 domain: all real numbers, range: y y ≥ 0

What are the domain and range of this function?
y=4x+1
domain: x x<0, range: y y<-1
domain: x x<0, range: y y ≥ 0
domain: all real numbers, range: y y<-1
domain: all real numbers, range: y y ≥ 0

Transcript text: What are the domain and range of this function? \[ y=4|x+1| \] domain: $\{x \mid x<0\}$, range: $\{y \mid y<-1\}$ domain: $\{x \mid x<0\}$, range: $\{y \mid y \geq 0\}$ domain: all real numbers, range: $\{y \mid y<-1\}$ domain: all real numbers, range: $\{y \mid y \geq 0\}$

Solution

Solution Steps

To determine the domain and range of the function $ y = 4|x+1| $, we need to consider the properties of absolute value functions. The domain of an absolute value function is all real numbers because you can input any real number into the function. The range is determined by the fact that the absolute value is always non-negative, and multiplying by 4 scales the output. Therefore, the range is all non-negative real numbers.

Step 1: Determine the Domain

The function $ y = 4|x + 1| $ is defined for all real numbers $ x $. Therefore, the domain is: \[ \text{Domain: } \mathbb{R} \]

Step 2: Determine the Range

The absolute value function $ |x + 1| $ is always non-negative, meaning $ |x + 1| \geq 0 $. When multiplied by 4, the output remains non-negative: \[ y = 4|x + 1| \geq 0 \] Thus, the range is: \[ \text{Range: } [0, \infty) \]