Question

when $x \leq 2$, what equation is shown in the graph?
$f(x)=|x - 1|$
$f(x)=3$
$f(x)=|x|$
$f(x)=x$

Explanation:

Step1: Analyze the graph's shape

The graph for \( x \leq 2 \) has a vertex at the origin \((0,0)\) and is a V - shaped graph, which is characteristic of an absolute - value function. The general form of an absolute - value function is \( y = |x| \), which has a vertex at \((0,0)\) and has two linear parts: \( y=-x \) for \( x < 0 \) and \( y = x \) for \( x\geq0 \), matching the shape of the given graph for \( x\leq2 \).

Let's check the other options:

• For \( f(x)=|x - 1| \), the vertex of the absolute - value function \( y = |x - h| \) is at \( (h,0) \). Here, \( h = 1 \), so the vertex should be at \( (1,0) \), which does not match the graph (the vertex is at \((0,0)\)).
• For \( f(x)=3 \), this is a horizontal line \( y = 3 \), which does not match the V - shaped graph.
• For \( f(x)=x \), this is a linear function with a slope of 1 and passes through the origin, but it does not have the V - shape (it is a straight line with a positive slope, not the two - part V - shape of the absolute - value function).

Step2: Conclusion

Based on the shape of the graph (V - shaped with vertex at the origin) and the analysis of the other functions, the function that represents the graph for \( x\leq2 \) is \( f(x)=|x| \).

Answer:

\( f(x)=|x| \) (the third option: \( f(x)=|x| \))