To determine which of the given functions is linear, we need to understand what defines a linear function. A linear function has the general form [tex]\( g(t) = at + b \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and the variable [tex]\( t \)[/tex] has the exponent of 1.
Let's analyze each function one by one:
1. [tex]\( g(t) = |3.4 t| + 9.8 \)[/tex]
- This function uses the absolute value of [tex]\( 3.4 t \)[/tex]. The absolute value function is not linear because it does not maintain the properties of linearity (additivity and homogeneity). Therefore, this function is not linear.
2. [tex]\( g(t) = 3.4 t^2 + 9.8 \)[/tex]
- This function includes a quadratic term [tex]\( t^2 \)[/tex]. A quadratic term involves the variable [tex]\( t \)[/tex] raised to the power of 2. Because linear functions must have the variable [tex]\( t \)[/tex] raised to the power of 1, this function is not linear.
3. [tex]\( g(t) = 3.4 t + 9.8 \)[/tex]
- This function follows the form [tex]\( g(t) = at + b \)[/tex], where [tex]\( a = 3.4 \)[/tex] and [tex]\( b = 9.8 \)[/tex]. The variable [tex]\( t \)[/tex] is raised to the power of 1. Therefore, this function fits the definition of a linear function, so it is linear.
4. [tex]\( g(t) = 3.4 t^3 + 9.8 \)[/tex]
- This function includes a cubic term [tex]\( t^3 \)[/tex]. A cubic term involves the variable [tex]\( t \)[/tex] raised to the power of 3. Since linear functions require the variable [tex]\( t \)[/tex] to be raised to the power of 1, this function is not linear.
Based on the analysis, the function that is linear is:
[tex]\( \boxed{3} \)[/tex] [tex]\( g(t) = 3.4 t + 9.8 \)[/tex].
