Question
Which graph is a function of x?
On a coordinate plane, a vertical line is at x = negative 3.
On a coordinate plane, a v-shaped graph opens up.
On a coordinate plane, a parabola opens to the right.
On a coordinate plane, one curve opens to the left and another curve opens to the right.
On a coordinate plane, a vertical line is at x = negative 3.
On a coordinate plane, a v-shaped graph opens up.
On a coordinate plane, a parabola opens to the right.
On a coordinate plane, one curve opens to the left and another curve opens to the right.
Answers
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Answered by
GPT-4o mini
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:
1. **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.
2. **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.
3. **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 \(x\).
4. **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**.
Let's analyze each of the options:
1. **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.
2. **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.
3. **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 \(x\).
4. **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**.
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