Question

What is the domain and range of the logarithmic function y=log2(x+3)+5
?(1 point)
Responses

Domain: (−∞,∞)
; Range: (−3,∞)
Domain: open paren negative infinity comma infinity close paren ; Range: open paren negative 3 comma infinity close paren

Domain: (5,∞)
; Range: (−∞,∞)
Domain: open paren 5 comma infinity close paren ; Range: open paren negative infinity comma infinity close paren

Domain: (−3,∞)
; Range: (−∞,∞)
Domain: open paren negative 3 comma infinity close paren ; Range: open paren negative infinity comma infinity close paren

Domain: (−3,∞)
; Range: [5,∞)

Answer 1

To determine the domain and range of the function \( y = \log_2(x + 3) + 5 \), we analyze the logarithmic part of the function.

### Domain:
The argument of the logarithm must be greater than zero. Therefore, we need:

\[
x + 3 > 0
\]

Solving for \( x \):

\[
x > -3
\]

Thus, the domain is:

\[
(-3, \infty)
\]

### Range:
The logarithmic function \( \log_2(x + 3) \) can take any real number value from \( -\infty \) to \( \infty \). Adding 5 shifts the entire graph upwards by 5 units. Therefore:

- The minimum value of \( y = \log_2(x + 3) + 5 \) occurs as \( x + 3 \) approaches 0 (making \( \log_2(x + 3) \) approach \( -\infty \)), thus \( y \) approaches \( -\infty \).
- The maximum value approaches \( \infty \) as \( x + 3 \) increases.

Thus, the range is:

\[
(-\infty, \infty)
\]

### Conclusion:
- **Domain**: \( (-3, \infty) \)
- **Range**: \( (-\infty, \infty) \)

**Correct answer:**
- Domain: (−3,∞); Range: (−∞,∞)