To determine which graph represents a function of \(x\), we can use the vertical line test. A graph represents a function if any vertical line drawn through the graph intersects it at most once.
Let's analyze each of the options:
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A vertical line at \( x = -3 \): This graph is not a function because a vertical line intersects all \(y\) values at \( x = -3\), failing the vertical line test.
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A v-shaped graph that opens up: This graph is a "V" shape which typically represents an absolute value function (like \(y = |x|\)). Any vertical line will intersect this graph at most once, which means it is a function.
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A parabola that opens to the right: Although a parabola that opens to the right is often expressed as a function of \(y\) (like \(x = y^2\)), it fails the vertical line test because a vertical line can intersect it at two points. Thus, it is not a function of \(x\).
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One curve opens to the left and another curve opens to the right: This description suggests a graph that might have two branches (like the hyperbola or a cubic function). Depending on the shape, it can potentially intersect a vertical line at more than one point, so it may or may not be a function.
Based on this analysis, the only clear function of \(x\) is:
- The v-shaped graph that opens up.
1 answer