To determine the type of function given by [tex]\( c(x) = 7^x - 2 \)[/tex], let's analyze its components.
1. Identify the structure of [tex]\( c(x) \)[/tex]:
- The function [tex]\( c(x) \)[/tex] is in the form [tex]\( 7^x - 2 \)[/tex].
- Notice that the variable [tex]\( x \)[/tex] is in the exponent.
2. Analyze the components:
- The base of the exponential expression is 7.
- The expression [tex]\( 7^x \)[/tex] is an exponential term because the variable [tex]\( x \)[/tex] appears as the exponent.
- Subtracting 2 from [tex]\( 7^x \)[/tex] is a transformation that shifts the graph of [tex]\( 7^x \)[/tex] downward by 2 units.
3. Characteristics of different functions:
- Linear Function: A linear function has the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants. The variable [tex]\( x \)[/tex] is of first degree, meaning it is not in an exponent or under any other transformation.
- Quadratic Function: A quadratic function has the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants. The variable [tex]\( x \)[/tex] is squared.
- Exponential Function: An exponential function has the form [tex]\( y = a \cdot b^x + c \)[/tex], where [tex]\( a \)[/tex] and [tex]\( c \)[/tex] are constants, and [tex]\( b \)[/tex] is the base of the exponent with the variable [tex]\( x \)[/tex].
4. Apply the characteristics:
- In [tex]\( c(x) = 7^x - 2 \)[/tex]:
- We see that [tex]\( 7 \)[/tex] is the base raised to the power of [tex]\( x \)[/tex], matching the form of [tex]\( b^x \)[/tex] in an exponential function.
- The subtraction of 2 is a constant term and shifts the exponential function vertically but does not change its fundamental nature.
Therefore, based on the analysis, the function [tex]\( c(x) = 7^x - 2 \)[/tex] is classified as an Exponential function.
