Sure, let's walk through the process of identifying and naming the type of each function step-by-step.
#### a) [tex]\( y = |x - 3| \)[/tex]
This function involves an absolute value operation on [tex]\( x - 3 \)[/tex]. The absolute value function transforms any input into its non-negative value.
- Type of Function: Absolute value
#### b) [tex]\( f(x) = 3(2)^x \)[/tex]
Here, we have a function where [tex]\( x \)[/tex] is the exponent of a base number (in this case, 2). Multiplying by a constant (3) does not change the nature of the function.
- Type of Function: Exponential
#### c) [tex]\( y = x^2 - 4 \)[/tex]
This function is in the form of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] where [tex]\( x \)[/tex] is raised to the power of 2 and constants are added or subtracted.
- Type of Function: Quadratic
#### d) [tex]\( g(x) = 4 \)[/tex]
This function is a constant function, where the value of the function does not change regardless of the input [tex]\( x \)[/tex].
- Type of Function: Linear (constant)
#### e) [tex]\( y = \frac{2}{3}x \)[/tex]
This function is in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m = \frac{2}{3} \)[/tex] and [tex]\( b = 0 \)[/tex]. This represents a straight line through the origin with a slope of [tex]\( \frac{2}{3} \)[/tex].
- Type of Function: Linear
To summarize:
- [tex]\( y = |x - 3| \)[/tex] is an Absolute value function.
- [tex]\( f(x) = 3(2)^x \)[/tex] is an Exponential function.
- [tex]\( y = x^2 - 4 \)[/tex] is a Quadratic function.
- [tex]\( g(x) = 4 \)[/tex] is a Linear (constant) function.
- [tex]\( y = \frac{2}{3} x \)[/tex] is a Linear function.
