find the value of $f(7)$.\n$y = f(x)$

Explanation:

Step1: Analyze the right - hand part of the function graph

The right - hand part of the function (y = f(x)) (the part where (x\geq1)) is a straight line. We can first find the equation of this line. We know two points on this line: when (x = 1), from the graph, the vertex of the triangle is at ((1,4)), and when (x = 5), (y=0) (since the graph intersects the (x) - axis at ((5,0))).

The slope (m) of a line passing through two points ((x_1,y_1)) and ((x_2,y_2)) is given by the formula (m=\frac{y_2 - y_1}{x_2 - x_1}). Let ((x_1,y_1)=(1,4)) and ((x_2,y_2)=(5,0)). Then (m=\frac{0 - 4}{5 - 1}=\frac{- 4}{4}=-1).

Using the point - slope form of a line (y - y_1=m(x - x_1)), with ((x_1,y_1)=(1,4)) and (m=-1), we get (y - 4=-1(x - 1)), which simplifies to (y=-x + 1+4=-x + 5).

Step2: Calculate (f(7))

To find (f(7)), we substitute (x = 7) into the equation of the right - hand line (y=-x + 5).

When (x = 7), (y=-7 + 5=-2). So (f(7)=-2).

Answer:

(-2)