Question

which statement is true about the two given functions?
a. f(x) and g(x) have the same value at x = -2
c. f(x) and g(x) have the same value at x = 2
d. f(x) and g(x) have the same value at x = 4
b. f(x) and g(x) have the same value at x = 0

Explanation:

Step1: Analyze the graph (assuming it's a piecewise linear function, likely an absolute - value - like graph)

First, we need to recall how to read the value of a function from its graph. For a given \(x\) - value, we find the corresponding \(y\) - value on the graph of the function. Let's assume the graph is of a function (let's say \(f(x)\) or \(g(x)\) or the combination). But from the options, we need to check each \(x\) - value:

- For \(x = - 2\): We look at the graph. If the graph has a point at \(x=-2\), we find its \(y\) - coordinate. Let's assume the left - hand part of the graph (the line with negative slope) passes through \(x = - 2\). Let's calculate the equation of the left - hand line. Suppose the vertex is at \((0, - 2)\) (from the graph's appearance, the lowest point is at \(x = 0,y=-2\)). The left - hand line: let's take two points. Let's say when \(x=-2\), if the slope of the left line is \(- 1\) (since from \(x=-2\) to \(x = 0\), the change in \(x\) is \(2\), and change in \(y\) is \(2\) (from \(y = 0\) to \(y=-2\)? Wait, maybe better to look at the grid. If each grid square is 1 unit. Let's assume the left line goes from \((-2,0)\) to \((0, - 2)\), so the slope is \(\frac{-2 - 0}{0-(-2)}=-1\). The equation of the left line is \(y-0=-1(x + 2)\), so \(y=-x - 2\). At \(x=-2\), \(y = 0\). The right - hand line: from \((0, - 2)\) to \((2,0)\) (slope \(\frac{0+2}{2 - 0}=1\)), equation \(y + 2=1(x - 0)\), \(y=x - 2\).

Now, for \(x=-2\), the left line gives \(y = 0\). Now, let's assume the other function (if there is another function, but maybe the graph is of \(f(x)\) and we are comparing with \(g(x)\), but since the options are about \(f(x)\) and \(g(x)\) having the same value at a point, let's check each option:

- Option A: \(x=-2\). From the left line, \(y = 0\). Let's assume the other function (maybe \(g(x)\) has the same value at \(x=-2\)). Wait, maybe the graph is of \(f(x)\) and we are to check the value. But let's check other options:

- Option B: \(x = 0\). At \(x = 0\), the \(y\) - value of the graph is \(-2\). If the other function has a different value at \(x = 0\), this is wrong.

- Option C: \(x = 2\). The right - hand line: \(y=x - 2\), when \(x = 2\), \(y=2 - 2=0\). If the other function has a different value at \(x = 2\), this is wrong.

- Option D: \(x = 4\). The right - hand line at \(x = 4\), \(y=4 - 2 = 2\). If the other function has a different value, this is wrong.

Wait, maybe the graph is of \(f(x)\) and we are to check the value. Let's re - evaluate. If the graph is \(y=\vert x\vert-2\)? No, when \(x = 0\), \(y=-2\), \(x = 2\), \(y = 0\), \(x=-2\), \(y = 0\). Ah! So the function \(y=\vert x\vert-2\) has at \(x=-2\), \(y=\vert - 2\vert-2=0\); at \(x = 2\), \(y=\vert2\vert-2 = 0\); at \(x = 0\), \(y=-2\); at \(x = 4\), \(y=\vert4\vert-2 = 2\).

Now, if we assume that \(f(x)\) and \(g(x)\) are such that at \(x=-2\), both have \(y = 0\), then Option A is correct. Wait, maybe the other function (if there is a table or another graph, but since we have only one graph, maybe the options are about the function in the graph and another function. But from the options, let's check:

If the graph is \(y=\vert x\vert-2\), then:

- At \(x=-2\), \(y = 0\)

- At \(x = 0\), \(y=-2\)

- At \(x = 2\), \(y = 0\)

- At \(x = 4\), \(y = 2\)

Now, if we assume that the two functions (maybe \(f(x)\) is the graph and \(g(x)\) is another function, but since the options are about same value, let's check which option is correct. Wait, maybe the original problem has a table for \(g(x)\), but since it's not provided, but from the graph's appearance, at \(x=-2\), the value is \(0\). So Option A says \(f(x)\) and \(g(x)\) have the same value at \(x=-2\), which is likely correct.

Step2: Eliminate other options

- Option B: At \(x = 0\), the graph has \(y=-2\). If the other function has a different value (e.g., maybe \(g(x)\) has a different value at \(x = 0\)), so B is wrong.

- Option C: At \(x = 2\), the graph has \(y = 0\), but maybe the other function has a different value, so C is wrong.

- Option D: At \(x = 4\), the graph has \(y = 2\), and the other function may have a different value, so D is wrong.

Answer:

A. \(f(x)\) and \(g(x)\) have the same value at \(x = - 2\)