the function below shows the portion of the even function f(x) for x≥0. sketch the portion of f(x) for x<0…

Explanation:

Step1: Recall property of even function

For an even function (f(x)=f( - x)). So the graph for (x < 0) is the reflection of the graph for (x\geq0) across the (y) - axis.

Step2: Recall average - rate - of - change formula

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here (a=-7) and (b = 5). First, since (f(x)) is even, (f(-7)=f(7)). From the graph, we need to estimate (f(5)) and (f(-7)=f(7)). Let's assume we can read the values of the function from the graph: (f(5)=y_1) and (f(-7)=f(7)=y_2). Then the average rate of change is (\frac{f(5)-f(-7)}{5-(-7)}=\frac{y_1 - y_2}{12}).

Step3: Find solutions of (f(x)=4)

To find the number of solutions of (f(x)=4), we draw the horizontal line (y = 4) on the entire graph (after sketching the part for (x<0) using the even - function property). Count the number of intersection points of (y = 4) and (y = f(x)).

Answer:

For sketching: Reflect the (x\geq0) part of the graph across the (y) - axis. For average rate of change: Calculate (\frac{f(5)-f(-7)}{12}) after reading (f(5)) and (f(-7)=f(7)) from the graph. For number of solutions of (f(x)=4): Count the intersection points of (y = 4) and the entire graph of (y = f(x)).