Question

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

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:

\(-2\)