Question

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

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