Question - Analyzing Functions: Domain, Range, and Functionality
Solution:
The task is to determine the domain, range, and whether each graph represents a function for each of the six examples.A function is defined as a relation where every input (usually x) has exactly one output (usually y). A common test for functions is the vertical line test. If a vertical line intersects a graph more than once, then the graph does not represent a function.a) For the first graph, which looks like a parabola facing upwards:- The domain is all real numbers since the parabola continues infinitely in both left and right directions. (Domain: ℝ or (-∞, ∞))- The range is all real numbers greater than or equal to the minimum value of the parabola. Since the vertex is at the x-axis, the minimum y-value is 0. (Range: [0, ∞))- The graph passes the vertical line test, hence it is a function.b) For the second graph, which is a semicircle:- The domain is the length of the semicircle's base which ranges from -5 to 5. (Domain: [-5, 5])- The range is the height of the semicircle which ranges from 0 to 5 or the radius of the semicircle. (Range: [0, 5])- The graph does not pass the vertical line test (any vertical line between x = -5 and x = 5 would intersect the graph twice), hence it is not a function.c) For the third graph, which appears to be a cubic polynomial:- The domain is all real numbers as the graph continues infinitely in both the left and right directions. (Domain: ℝ or (-∞, ∞))- The range is also all real numbers, as the graph goes infinitely in the upward and downward directions. (Range: ℝ or (-∞, ∞))- The graph passes the vertical line test, hence it is a function.d) For the fourth graph, which looks like an absolute value function reflected over the x-axis:- The domain is all real numbers as the graph continues infinitely both to the left and right. (Domain: ℝ or (-∞, ∞))- The range includes all real numbers less than or equal to the maximum value at the x-axis. Since the vertex is at the x-axis, the maximum y-value is 0. (Range: (-∞, 0])- The graph passes the vertical line test, hence it is a function.e) For the fifth graph, which is a straight line with a negative slope:- The domain is all real numbers, as the line extends infinitely in both left and right directions. (Domain: ℝ or (-∞, ∞))- The range is also all real numbers, as the line extends infinitely in both the upward and downward directions. (Range: ℝ or (-∞, ∞))- The graph passes the vertical line test, hence it is a function.f) For the sixth graph, which looks like an arrow going downward:- The domain is all real x-values where the graph is defined. Since the graph is an arrow pointing downwards and stops, the domain is not clearly defined from the image, but it might stop at x = 6 as it appears to. A reasonable assumption would be that domain is all real numbers less than or equal to 6. (Domain: (-∞, 6] or another interval depending on where the graph actually ends.)- The range is the set of y-values that the arrow points to. In this case, it appears to point infinitely downwards, so the range would be all real numbers. (Range: ℝ or (-∞, ∞))- The graph does not pass the vertical line test since, at x = 6, it appears that multiple y-values are associated with this single x-value. However, since the arrow might indicate a single limit point, without more context or clarity about the graph's intention, it's ambiguous whether this is a function. If the arrow indicates a single point at x = 6, it could still be considered a function. But if the arrow indicates an interval, then it's not a function.
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