Question
Find the domain and range for the following function
y = |x|+4
Domain: (-00,00) Range: (-00,00)
Domain: (4,∞) Range: (-00,00)
Domain: (4,∞) Range: (-00,00)
Domain: (-∞,∞) Range: (4,0∞)
y = |x|+4
Domain: (-00,00) Range: (-00,00)
Domain: (4,∞) Range: (-00,00)
Domain: (4,∞) Range: (-00,00)
Domain: (-∞,∞) Range: (4,0∞)
Answers
GPT-4o mini
To find the domain and range of the function \( y = |x| + 4 \), let's analyze it step by step.
### Domain:
The domain of a function is the set of all possible input values (x-values) that the function can accept. In this case, the function \( y = |x| + 4 \) can accept any real number for \( x \), since the absolute value function is defined for all real numbers.
Thus, the domain is:
\[
\text{Domain: } (-\infty, \infty)
\]
### Range:
The range of a function is the set of all possible output values (y-values) of the function. The expression \( |x| \) is always non-negative (i.e., \( |x| \geq 0 \)). Therefore, \( |x| + 4 \) will be at least 4 (when \( x = 0\)).
As \( x \) approaches positive or negative infinity, \( |x| \) also approaches infinity, so \( |x| + 4 \) will approach infinity as well.
Therefore, the minimum value of \( y \) is 4, and there is no upper limit on \( y \).
Thus, the range is:
\[
\text{Range: } [4, \infty)
\]
In conclusion, the correct domain and range for the function \( y = |x| + 4 \) are:
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [4, \infty) \)
**Note:** The provided options enclosed in parentheses like \((4, \infty)\) and \((-\infty, \infty)\) were misrepresentations of the range. The correct representation would include \(4\), hence in interval notation it is \([4, \infty)\).
### Domain:
The domain of a function is the set of all possible input values (x-values) that the function can accept. In this case, the function \( y = |x| + 4 \) can accept any real number for \( x \), since the absolute value function is defined for all real numbers.
Thus, the domain is:
\[
\text{Domain: } (-\infty, \infty)
\]
### Range:
The range of a function is the set of all possible output values (y-values) of the function. The expression \( |x| \) is always non-negative (i.e., \( |x| \geq 0 \)). Therefore, \( |x| + 4 \) will be at least 4 (when \( x = 0\)).
As \( x \) approaches positive or negative infinity, \( |x| \) also approaches infinity, so \( |x| + 4 \) will approach infinity as well.
Therefore, the minimum value of \( y \) is 4, and there is no upper limit on \( y \).
Thus, the range is:
\[
\text{Range: } [4, \infty)
\]
In conclusion, the correct domain and range for the function \( y = |x| + 4 \) are:
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [4, \infty) \)
**Note:** The provided options enclosed in parentheses like \((4, \infty)\) and \((-\infty, \infty)\) were misrepresentations of the range. The correct representation would include \(4\), hence in interval notation it is \([4, \infty)\).