Question

the average rate of change of ( g(x) ) between ( x = 4 ) and ( x = 7 ) is ( \frac{5}{6} ). which statement must be true?
( g(7)-g(4)=\frac{5}{6} )
( \frac{g(7 - 4)}{7 - 4}=\frac{5}{6} )
( \frac{g(7)-g(4)}{7 - 4}=\frac{5}{6} )
( \frac{g(7)}{g(4)}=\frac{5}{6} )

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = g(x)\) over the interval \([a,b]\) is given by \(\frac{g(b)-g(a)}{b - a}\). Here, \(a = 4\) and \(b=7\).

Step2: Substitute the values into the formula

Substituting \(a = 4\) and \(b = 7\) into the formula \(\frac{g(b)-g(a)}{b - a}\), we get \(\frac{g(7)-g(4)}{7 - 4}\). Since the average rate of change is \(\frac{5}{6}\), we have \(\frac{g(7)-g(4)}{7 - 4}=\frac{5}{6}\).

Answer:

\(\frac{g(7)-g(4)}{7 - 4}=\frac{5}{6}\) (the third option)