To determine whether the given function [tex]\( y = 9x - 8 \)[/tex] is an exponential function, we need to understand the definitions of linear and exponential functions.
1. Linear Functions: A linear function can be written in the form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept.
In this form, [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex] changes. There are no exponential terms (e.g., terms where [tex]\( x \)[/tex] is an exponent).
2. Exponential Functions: An exponential function can be written in the form [tex]\( y = a \cdot b^x \)[/tex], where:
- [tex]\( a \)[/tex] is a constant.
- [tex]\( b \)[/tex] is the base of the exponential term and is also a constant.
In this form, [tex]\( y \)[/tex] changes multiplicatively as [tex]\( x \)[/tex] changes. The variable [tex]\( x \)[/tex] appears in the exponent.
Now, evaluate the given function [tex]\( y = 9x - 8 \)[/tex]:
- Compare it to the linear form [tex]\( y = mx + b \)[/tex]:
- Here, [tex]\( m = 9 \)[/tex]
- and [tex]\( b = -8 \)[/tex]
Since [tex]\( y = 9x - 8 \)[/tex] fits the linear form, it is a linear function.
- Compare it to the exponential form [tex]\( y = a \cdot b^x \)[/tex]:
- The function does not have an exponential term where [tex]\( x \)[/tex] is an exponent.
Therefore, the given function [tex]\( y = 9x - 8 \)[/tex] is not an exponential function.
The function [tex]\( y = 9x - 8 \)[/tex] is classified as a linear function.
Thus, the correct answer is:
- Not exponential
