Question

which equation represents the function?
the image shows a graph of a function (a v - shaped graph, likely an absolute - value function) and four options:
h(x)=|x + 1|
f(x)=|x|−1
g(x)=|x−1|
j(x)=|x| + 1

Explanation:

Step1: Analyze the vertex of the absolute - value graph

The general form of an absolute - value function is \(y = |x - h|+k\), where \((h,k)\) is the vertex of the graph. From the given graph, we can see that the vertex is at the point \((1,0)\).

Step2: Substitute the vertex into the general form

For the function \(y = |x - h|+k\), substituting \(h = 1\) and \(k = 0\) into the formula, we get \(y=|x - 1|\), which is the function \(g(x)=|x - 1|\). We can also verify by checking the value of the function at \(x = 0\). For \(g(x)=|x - 1|\), when \(x = 0\), \(g(0)=|0 - 1|=1\)? Wait, no, wait the graph: Wait, let's re - examine the graph. Wait, the vertex is at \((1,0)\). Let's check the function \(g(x)=|x - 1|\): when \(x = 1\), \(g(1)=|1 - 1| = 0\) (correct vertex). When \(x=0\), \(g(0)=|0 - 1|=1\)? But the graph at \(x = 0\) has a \(y\) - value of 0? Wait, no, wait the graph: Wait, the left - hand side of the graph: let's take a point on the left - hand line. For example, when \(x=- 1\), what's the \(y\) - value? Wait, no, the vertex is at \((1,0)\). Let's check the function \(g(x)=|x - 1|\): when \(x = 2\), \(g(2)=|2 - 1| = 1\); when \(x=3\), \(g(3)=|3 - 1| = 2\), which matches the right - hand side of the V - shaped graph. For the left - hand side, when \(x = 0\), \(g(0)=|0 - 1|=1\)? Wait, no, maybe I made a mistake. Wait, let's look at the other functions:

• For \(h(x)=|x + 1|\), the vertex is at \((-1,0)\), which does not match the vertex \((1,0)\) of the given graph.
• For \(f(x)=|x|-1\), the vertex is at \((0, - 1)\), which does not match the vertex \((1,0)\).
• For \(j(x)=|x| + 1\), the vertex is at \((0,1)\), which does not match the vertex \((1,0)\).
• For \(g(x)=|x - 1|\), the vertex is at \((1,0)\), which matches the vertex of the given graph.

Answer:

\(g(x)=|x - 1|\) (or the option corresponding to \(g(x)=|x - 1|\))