Question

problem 3

which equation represents the function?
select the correct answer choice.

a ( f(x) = |x| - 1 )
b ( g(x) = |x - 1| )
c ( h(x) = |x + 1| )
d ( j(x) = |x| + 1 )

(graph: a v - shaped graph with vertex at (0, 0) shifted? wait, looking at the graph, the vertex is at (0, 0)? wait no, the graph shown has vertex at (0, 0)? wait the y - axis: when x = 0, y = 0? wait the original graph (from the image) has the vertex at (0, 0)? wait the options are absolute value functions. lets re - ocr the text properly: the problem is about a graph of an absolute - value function, with options a to d as above, and the question is which equation represents the function? select the correct answer choice.

Explanation:

Step1: Analyze the vertex of the absolute - value function

The general form of an absolute - value function is \(y = |x - h|+k\), where \((h,k)\) is the vertex of the V - shaped graph. From the given graph, the vertex of the function is at \((0, - 1)\)? Wait, no, looking at the graph, when \(x = 0\), \(y=- 1\)? Wait, no, let's check the graph again. Wait, the graph passes through \((0, - 1)\)? Wait, no, the graph has its vertex at \((0,-1)\)? Wait, no, let's take a point. When \(x = 0\), the \(y\) - value is \(-1\)? Wait, no, the graph: when \(x = 0\), the point is \((0,-1)\)? Wait, no, looking at the grid, the vertex is at \((0, - 1)\)? Wait, no, the original absolute - value function \(y = |x|\) has a vertex at \((0,0)\). The given graph has a vertex at \((0,-1)\). The function \(y=|x|-1\) is a vertical shift of the function \(y = |x|\) down by 1 unit. Let's check the options:

• Option A: \(f(x)=|x|-1\). The vertex of \(y = |x|-1\) is at \((0,-1)\), which matches the vertex of the given graph.
• Option B: \(g(x)=|x - 1|\). The vertex of \(y=|x - 1|\) is at \((1,0)\), which does not match the vertex of the given graph.
• Option C: \(h(x)=|x + 1|\). The vertex of \(y=|x + 1|\) is at \((-1,0)\), which does not match the vertex of the given graph.
• Option D: \(j(x)=|x|+1\). The vertex of \(y = |x|+1\) is at \((0,1)\), which does not match the vertex of the given graph.

We can also check by plugging in \(x = 0\) into each function:

• For option A: \(f(0)=|0|-1=0 - 1=-1\), which matches the \(y\) - value when \(x = 0\) in the graph.
• For option B: \(g(0)=|0 - 1|=| - 1| = 1

eq-1\)

• For option C: \(h(0)=|0 + 1|=|1| = 1

eq-1\)

• For option D: \(j(0)=|0|+1=0 + 1 = 1

eq-1\)

Answer:

A. \(f(x)=|x|-1\)