To determine the family to which the function [tex]\( y = 2^x + 5 \)[/tex] belongs, let's carefully analyze its form and characteristics.
1. Identify the general form of the function:
The given function is [tex]\( y = 2^x + 5 \)[/tex].
2. Compare with known function families:
- Quadratic function:
A quadratic function has the general form [tex]\( y = ax^2 + bx + c \)[/tex]. The given function does not match this form because it does not include the term [tex]\( x^2 \)[/tex].
- Square root function:
A square root function has the general form [tex]\( y = a\sqrt{x} + b \)[/tex]. The given function does not include a square root term.
- Exponential function:
An exponential function has the general form [tex]\( y = a \cdot b^x + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( b \)[/tex] is a positive real number other than 1. The given function, [tex]\( y = 2^x + 5 \)[/tex], matches this form since we can identify [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 5 \)[/tex].
- Logarithmic function:
A logarithmic function has the general form [tex]\( y = a \cdot \log_b(x) + c \)[/tex]. The given function does not match this form because it does not include a logarithmic term.
After comparing the function with all the families, it is clear that [tex]\( y = 2^x + 5 \)[/tex] belongs to the exponential family.
Hence, the function [tex]\( y = 2^x + 5 \)[/tex] is an exponential function.
