Linear functions can be represented by a straight line, with an intercept and a constant slope. They are formed by a dependent varible (y) and an independent varible (x) whose power equals one, which means that y is related to x in a linear way (they have both power equal to one).
The general equation of a linear function can be written as follows: y= a + b x, with a equal to a constant known as the intercept, and b equal to the slope.
We can rewrite the equations you have attached as follow, by rearranging terms in order to clear y as a function of x (in order):
[tex]y=-19+\frac{1}{2}x[/tex], is a linear function: y is linked to x in a linear way, the curve has the form y= a + b x, where a=-19 and b=1/2.
[tex]y=\frac{1}{3} x^{2}[/tex], it is NOT a linear function, because y is linked to x in a way that is not linear (the power of x is different from zero, in this case the power of x equals 2 ), and the graph of this expression would not be linear.
[tex]y=\frac{1}{39}x +\frac{5}{13}[/tex] is a linear function: y is linked to x in a linear way (the power of x equals one), the curve in this case has the form y= a + b x, where a=1/39 and b=5/13.
[tex]y=x+\frac{25}{5}[/tex] is a linear function: y is linked to x in a linear way (the power of x equals one), and the curve has the form y= a + b x, where a=25/5 and b=1.
[tex]y=\sqrt[3]{x}[/tex] is NOT a linear function, because y is linked to x in a nonlinear way, specifically, y is linked to the squared root of x, which means that is linked to [tex]x^{\frac{1}{3} }[/tex], then x has not power equal to one in this case.
