This question (and the next four) should give you a flavor for what learning by play and experiment is like! The topics here are all covered more thoroughly in this course.

In the function \( f(x) = ax^n ,\) as \(x\) increases starting at 0, under what circumstances is \( f(x) \) decreasing? (That is, when you look at the output of the function as the graph goes to the right, when does it go down rather than up?)

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In the visualization below, you can move the blue point anywhere on the grid. A second point called the *inverse* of that point moves along with it. What is the inverse of a coordinate point \( (x,y) ?\)

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The type of function in the visualization below is called a third-degree polynomial \((\)in the cases where \( a \neq 0 ).\)

**True or False?**

If the function \( f(x) \) is a third-degree polynomial, \( f(x) = 0 \) will always have at least one solution.

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A third-degree polynomial is able to be *symmetrical about the origin*. In such a case, if the graph is rotated 180 degrees using \( (0,0) \) as the center, an identical graph results. (That is, if you turn it upside down, nothing changes.)

Using the variables below, what needs to be true for this to be the case?

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The function \[ f(x) = \frac{1}{2x + a} \] is a rational function where the domain (the set of all valid inputs) is every real number except for one particular missing value. (That is, there is always a single value \(x\) that isn't a valid input.)

If we want \(x=12\) to be the value missing from the domain, what should \(a\) be?

(Hint: In the visualization, try to figure out where the "gap" is in relation to \(a.)\)

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