Question

the graph of $y = f(x)$ is shown below. find all values of $x$ for which $f(x) > 0$.

Explanation:

Step1: Identify x-intercepts

Find where the graph crosses the x - axis. From the graph, the x - intercepts are at \(x=-4\) and \(x = 0\) (wait, re - examining, maybe the upper curve crosses x - axis at \(x=-4\) and another point? Wait, the graph: the upper curve (a parabola - like) and a line? Wait, no, looking at the graph, the function \(y = f(x)\) has two x - intercepts? Wait, no, the graph shows that the curve (the upper one) intersects the x - axis at \(x=-4\) and maybe another point? Wait, no, the line and the curve? Wait, actually, the graph of \(y = f(x)\) (the upper curve) is a parabola opening to the left? Wait, no, the axes: x - axis is vertical? Wait, no, standard axes: x - axis horizontal, y - axis vertical. Wait, the labels: the horizontal axis is y? Wait, no, the right - hand side says "The graph of \(y = f(x)\) is shown below. Find all values of \(x\) for which \(f(x)>0\)". So the vertical axis is x? Wait, no, that's non - standard. Wait, maybe the horizontal axis is x and vertical is y. Wait, the graph: the upper curve (a parabola) and a line. Wait, the x - intercepts (where \(y = 0\)): looking at the graph, the upper curve (the one that is a "U" - shaped but opening to the left? No, maybe it's a parabola opening downward? Wait, no, the axes: the vertical axis is x (since the right - hand side is about x - values). Wait, this is a bit confusing. Wait, re - interpreting: the vertical axis is x, horizontal is y. So we need to find x (vertical) where \(f(x)>0\) (i.e., \(y>0\) for the function \(y = f(x)\), but x is the input. Wait, no, standard: \(y = f(x)\), x is horizontal, y is vertical. So the graph has x - axis horizontal, y - axis vertical. The function \(y = f(x)\) has two parts? No, the graph shows a curve (maybe a parabola) and a line. Wait, the x - intercepts (where \(y = 0\)): from the graph, the curve crosses the x - axis at \(x=-4\) and \(x = 0\)? Wait, no, looking at the graph, the upper curve (the one that is above the x - axis between two points) – wait, no, the graph: the horizontal axis is y, vertical is x? Wait, the right - hand text: "Find all values of x for which \(f(x)>0\)". So x is the input, y = f(x) is the output. So the vertical axis is y, horizontal is x. So the graph: the function \(y = f(x)\) has a curve (maybe a parabola) and a line. The x - intercepts (where \(y = 0\)) are at \(x=-4\) and \(x = 0\)? Wait, no, the graph: the upper curve (the one that is a "U" - like but opening to the left? No, maybe it's a parabola opening to the left, with vertex at (x, y)=( - 2,7) or something. Wait, no, let's look at the regions where \(y>0\) (since \(f(x)=y\), so \(f(x)>0\) means \(y>0\)). The graph: the upper curve (the non - linear part) is above the x - axis (y = 0) between two x - values? Wait, no, the x - axis is horizontal. Wait, the graph shows that the function \(y = f(x)\) (the upper curve) is above the x - axis (y>0) when x is between - 4 and 0? Wait, no, that can't be. Wait, maybe the horizontal axis is x, vertical is y. The upper curve (a parabola) has x - intercepts at \(x=-4\) and \(x = 0\), and is above the x - axis (y>0) between \(x=-4\) and \(x = 0\)? Wait, no, the parabola opening to the left: if the vertex is at (x, y)=( - 2,7), then the parabola \(y = a(x + 2)^2+7\), opening to the left (so x is a function of y). But we need \(f(x)>0\), i.e., y>0 for the function \(x = f(y)\)? No, the problem says "y = f(x)", so x is horizontal, y is vertical. I think there's a mislabeling in the graph, but assuming the standard axes (x horizontal, y vertical), the function \(y = f(x)\) has a curve (maybe a parabola) and a line. The x - intercepts (where \(y = 0\)) are at \(x=-4\) and \(x = 0\), and the curve is above the x - axis (y>0) between \(x=-4\) and \(x = 0\). Wait, no, the line: the lower line, but the upper curve is the one we care about? Wait, no, the problem says "the graph of \(y = f(x)\)", so it's the entire graph. Wait, the graph has two parts: a line and a curve. But when \(f(x)>0\), we look for where the graph is above the x - axis (y>0). From the graph, the region where \(y>0\) is between \(x=-4\) and \(x = 0\) (the upper curve is above the x - axis there), and also, wait, the line: is the line above the x - axis? No, the line is below? Wait, no, re - examining: the upper curve (the one that is a "U" - shaped) is above the x - axis between \(x=-4\) and \(x = 0\), and the line is below? Wait, no, the vertical axis is x? I think there's a mix - up. Wait, the correct way: for \(y = f(x)\), \(f(x)>0\) means \(y>0\). So we find the x - values where the graph is above the x - axis (y - axis horizontal? No, x - axis horizontal, y - axis vertical). So the graph: the upper curve (a parabola) intersects the x - axis at \(x=-4\) and \(x = 0\), and is above the x - axis (y>0) between \(x=-4\) and \(x = 0\). Wait, no, if the parabola opens to the left, then x is a function of y, but the problem is about x as the input. I think the graph is mis - axis - labeled, and the horizontal axis is x, vertical is y. So the x - intercepts (where \(y = 0\)) are at \(x=-4\) and \(x = 0\), and the function \(y = f(x)\) is positive (y>0) between \(x=-4\) and \(x = 0\).

Step2: Determine the interval

The graph of \(y = f(x)\) is above the x - axis (where \(y = 0\)) when \(x\) is between - 4 and 0, i.e., \(-4<x<0\).

Answer:

\(-4 < x < 0\)