Question

which of the following represents a function?
a rule that gives two outputs for one input
a graph that fails the vertical line test
a table with the same x - value paired to two different y - values
a set of points where no x - values repeat
question 2
what does it mean if a graph of a function is decreasing over an interval?
the y - values get smaller as x increases
the slope of the line is positive
the x - values decrease as y increases
the graph curves upward

Explanation:

Question 1:

Step 1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value).

• For a rule that gives two outputs for one input: This violates the definition of a function.
• For a graph that fails the vertical line test: The vertical line test states that if a vertical line intersects the graph at more than one point, it is not a function. So, a graph that fails the vertical line test is not a function.
• For a table with the same x - value paired to two different y - values: This also violates the function definition.
• For a set of points where no x - values repeat: Since each x - value has a unique y - value (because x - values don't repeat), this satisfies the function definition.

Question 2:

Step 1: Recall the concept of a decreasing function

A function \(y = f(x)\) is decreasing over an interval if for any two points \(x_1\) and \(x_2\) in the interval with \(x_1

• The slope of the line is positive: A positive - slope indicates an increasing function (\(y\) increases as \(x\) increases).
• The \(x\) - values decrease as \(y\) increases: This is not the standard definition of a decreasing function. The focus is on the behavior of \(y\) with respect to the increase of \(x\).
• The graph curves upward: This is related to concavity (for a twice - differentiable function, if \(y''>0\) the graph curves upward), not to the function being decreasing.

Answer:

1. A set of points where no \(x\) - values repeat
2. The \(y\) - values get smaller as \(x\) increases