To determine whether the function [tex]\( f(x) = 9^{x-1} + 2 \)[/tex] is linear, quadratic, or exponential, let's analyze its form and properties.
1. Linear Function:
- A linear function has the general form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Linear functions create a straight line when graphed and have a constant rate of change.
2. Quadratic Function:
- A quadratic function has the general form [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, with [tex]\( a \neq 0 \)[/tex].
- Quadratic functions create a parabolic curve when graphed and have a variable rate of change.
3. Exponential Function:
- An exponential function has the general form [tex]\( f(x) = a \cdot b^x + c \)[/tex], where [tex]\( a \)[/tex] and [tex]\( c \)[/tex] are constants, and [tex]\( b \)[/tex] is the base of the exponential term.
- Exponential functions involve a constant multiplicative rate of change.
Given the function [tex]\( f(x) = 9^{x-1} + 2 \)[/tex]:
- We have an expression that includes an exponential term [tex]\( 9^{x-1} \)[/tex].
- This term can be rewritten for clarity using properties of exponents: [tex]\( 9^{x-1} = 9^x \cdot 9^{-1} = \frac{9^x}{9} \)[/tex].
- We then add a constant term [tex]\( 2 \)[/tex].
This function fits the form of an exponential function [tex]\( a \cdot b^x + c \)[/tex], where [tex]\( a = \frac{1}{9} \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = 2 \)[/tex].
Thus, the function [tex]\( f(x) = 9^{x-1} + 2 \)[/tex] is an exponential function.
