Question

Determine the domain and range y = 5 * (1.5) ^ x + 2
Domain:
[ Select ]
all real numbers
x>1
x<1
0<x<1
Range:
(select)
all real numbers
y<2
y>2
-100<y<100

Answer 1

To determine the domain and range of the function \( y = 5 \cdot (1.5)^x + 2 \):

1. **Domain**:
The function \( 5 \cdot (1.5)^x + 2 \) is an exponential function, which is defined for all real numbers \( x \). Therefore, the domain is:
- **Domain: all real numbers**

2. **Range**:
The expression \( (1.5)^x \) is always positive for all real \( x \), and it approaches 0 as \( x \) approaches negative infinity. Thus, the minimum value of \( 5 \cdot (1.5)^x \) is 0 (as \( x \to -\infty \)), leading to a minimum value of \( y \) approaching 2:
- As \( x \to \infty \), \( (1.5)^x \) approaches infinity, and consequently, \( y \) approaches infinity as well.
- Therefore, \( y \) can take any value greater than 2.

Consequently, the range is:
- **Range: \( y > 2 \)**

So, the final answers would be:
- **Domain**: all real numbers
- **Range**: \( y > 2 \)