state whether the following statement is true or false. the function ( f(x)=\frac{2}{3}x + 15 ) is…

state whether the following statement is true or false. the function ( f(x)=\frac{2}{3}x + 15 ) is increasing on the interval ( (-infty,infty) ). choose the correct answer below. false true

state whether the following statement is true or false. the function ( f(x)=\frac{2}{3}x + 15 ) is increasing on the interval ( (-infty,infty) ). choose the correct answer below. false true

Answer

Explanation:

Step1: Recall the formula for the derivative of a linear function

The derivative of a function (y = mx + b) (where (m) is the slope and (b) is the y - intercept) is given by (y^\prime=m). For the function (f(x)=\frac{2}{3}x + 15), using the power rule ((x^n)^\prime=nx^{n - 1}), when (n = 1) (since (x=x^1)) and the derivative of a constant (b) is (0). So (f^\prime(x)=\frac{2}{3}).

Step2: Use the increasing - function criterion

A function (y = f(x)) is increasing on an interval (I) if (f^\prime(x)>0) for all (x\in I). Here, (f^\prime(x)=\frac{2}{3}>0) and the domain of (f(x)) (the set of all real numbers, since it is a linear function) is ((-\infty,\infty)).

Answer:

True