Exam One Review

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This is a (lengthy) practice exam for Exam 1.

NOTE: There may be random lines that look like \texttt{(SOME TEXT)} and maybe even some { } within that text. You can ignore everything that isn’t the text itself; ie the texttt bit and the braces, they are an artifact of how some randomization is being done that I haven’t had time to fix yet. If you literally delete the \texttt and the braces from what you see, and just keep the text itself, it will read exactly as intended. So just ignore all those texttt and braces whenever you see them.

Also note: Some of the below has randomized elements, some do not. Currently Xronos does not support randomized graphing (although we’re working on it!) so a lot of the graphing problems won’t randomize, but a surprising amount of the other problems will change values or text if you hit the green “another” button in the top right corner to get another version of this practice exam. This also means it may take some time for the entire test to render because we have to rely on public servers to do the randomization for us currently, so please be patient. If it takes more than 3-5 minutes to fill out the random values (meaning: if there are still spinning wheels of death going after 3-5 minutes) there is a problem. Try hitting the “another” button to see if it resolves itself. If it keeps doing this, please contact your instructor and let him know!

Suppose $f(x) = \sage {algHfunc1}$ , $g(x) = \sage {algHfunc2}$ , and $h(x) = \sage {algHfunc3}$ ; compute $(f\cdot (g-h))(\sage {algHc1})$ .

$\sage {algHans2}$ $\sage {algHans5}$ $\sage {algHans1}$ $\sage {algHans4}$ $\sage {algHans3}$

Suppose $f(x) = \sage {compEfunc1}$ and $g(x) = \sage {compEfunc2}$ ; compute $(f \circ g)(\sage {compEc1})$ .

$\sage {compEans4}$ $\sage {compEans5}$ $\sage {compEans3}$ $\sage {compEans2}$ $\sage {compEans1}$

Suppose $f(x) = \sage {p1algEfunc1}$ , $g(x) = \sage {p1algEfunc2}$ ; compute $(f - g)(\sage {p1algEc1})$ .

$\sage {p1algEans1}$ $\sage {p1algEans3}$ $\sage {p1algEans2}$ $\sage {p1algEans4}$ $\sage {p1algEans5}$

Suppose $f(x) = \sage {p2algEfunc1}$ , $g(x) = \sage {p2algEfunc2}$ ; compute $(f - g)(\sage {p2algEc1})$ .

$\sage {p2algEans4}$ $\sage {p2algEans5}$ $\sage {p2algEans3}$ $\sage {p2algEans2}$ $\sage {p2algEans1}$

Given: $f(\sage {p1algMx1})=\sage {p1algMy1}$ , $f(\sage {p1algMx2})=\sage {p1algMy2}$ , $g(\sage {p1algMx2})=\sage {p1algMy3}$ , $g(\sage {p1algMx3})=\sage {p1algMx1}$ Compute:

a.: $(fg)(\sage {p1algMx2})$
b.: $(f\circ g)(\sage {p1algMx3})$
c.: $(f-g)(\sage {p1algMx2})$

$(fg)(\sage {p1algMx2}) = \sage {p1algMans2}$ , $(f \circ g)(\sage {p1algMx3}) = \sage {p1algMans2}$ , $(f-g)(\sage {p1algMx2}) = \sage {p1algMans3}$ $(fg)(\sage {p1algMx2}) = \sage {p1algMans2}$ , $(f \circ g)(\sage {p1algMx3}) = \sage {p1algMans2}$ , $(f-g)(\sage {p1algMx2}) = \sage {p1algMans4}$ $(fg)(\sage {p1algMx2}) = \sage {p1algMans2}$ , $(f \circ g)(\sage {p1algMx3}) = \sage {p1algMans3}$ , $(f-g)(\sage {p1algMx2}) = \sage {p1algMans3}$ $(fg)(\sage {p1algMx2}) = \sage {p1algMans1}$ , $(f \circ g)(\sage {p1algMx3}) = \sage {p1algMans2}$ , $(f-g)(\sage {p1algMx2}) = \sage {p1algMans3}$ $(fg)(\sage {p1algMx2}) = \sage {p1algMans1}$ , $(f \circ g)(\sage {p1algMx3}) = \sage {p1algMans2}$ , $(f-g)(\sage {p1algMx2}) = \sage {p1algMans4}$

Given: $f(x) = \sage {p2algMf1}$ and $g(x) = \sage {x +p2algMc2}$ . Which of the following is equivalent to $(g\circ f)(\sage {x-p2algMc1})$

$\sage {p2algMans1}$ $\sage {p2algMansfalse1}$ $\sage {p2algMansfalse3}$ $\sage {p2algMansfalse4}$ $\sage {p2algMansfalse2}$

If a function is invertible, then $f$ and $f^{-1}$ must have the same parent function.

This is false, but only because $f^{-1}$ is invertible.This is true even for partial inverses.This would be true, but only for (all) partial inverse functions.False, it is possible for a function and its inverse to have different parent functions.True, but only if $f^{-1}$ is actually a true inverse.

If the point $(\sage {p1inverseHc1},\sage {p1inverseHc2})$ is on the graph of $f(x)$ , then which point is on the graph of $g^{-1}(x)$ (the graph of $g$ inverse) where $g(x) = \sage {p1inverseHc5}f(\sage {p1inverseHf1}) + (\sage {p1inverseHc6})$ ?

$(\sage {p1inverseHx},\sage {p1inverseHy})$ $(\sage {p1inverseHy},\sage {p1inverseHx})$ $(\sage {p1inverseHxfalse},\sage {p1inverseHyfalse})$ $(\sage {p1inverseHyfalse},\sage {p1inverseHxfalse})$ $(\sage {p1inverseHxfalse},\sage {p1inverseHy})$

Let $f(x)$ be a relation. If $f^{-1}(x)$ is a function, must it be true that $f(x)$ is a function?

Yes, $f^{-1}(x)$ being a function means it passes the horizontal line test, so $f(x)$ is therefore a function.No, if the inverse is a function, then $f(x)$ cannot be a function.Yes, because the inverse can only be defined if $f(x)$ is a function.Yes, $f^{-1}(x)$ being a function means it passes the vertical line test, so $f(x)$ is therefore a function.No, it’s possible the inverse is a function but the original would not be.

If $f^{-1}(x)$ exists, then $f(x)$ passes the horizontal line test.

This is only true for continuous functionsThis is never true.This is true, whether or not $f^{-1}(x)$ is a function.This is only true if $f^{-1}(x)$ is a function.There is not enough information to know if this is true.

Suppose, for some function $f(x)$ we have that $f(\sage {p1inverseEc1}) = \sage {p1inverseEc2}$ . What can we say about $f^{-1}(\sage {p1inverseEc2})$ ?

$f^{-1}(\sage {p1inverseEc2})$ must be $\sage {-p1inverseEc1}$ $f^{-1}(\sage {p1inverseEc2})$ exists, but the value is unknown. $f^{-1}(\sage {p1inverseEc2})$ definitely does not exist.There is not enough information to know if $f^{-1}(\sage {p1inverseEc2})$ exists. $f^{-1}(\sage {p1inverseEc2})$ must be $\sage {p1inverseEc1}$ .

If the point $(\sage {p2inverseEc1}, \sage {p2inverseEc2})$ is on the graph of $g(x)$ . Which of the following points must be on the graph of $g^{-1}(x)$ ?

$(\sage {-p2inverseEc2},\sage {-p2inverseEc1})$ $(\sage {p2inverseEc2},\sage {p2inverseEc1})$ $(\sage {-p2inverseEc1},\sage {-p2inverseEc2})$ $(\sage {p2inverseEc1},\sage {-p2inverseEc2})$ $(\sage {p2inverseEc2},\sage {-p2inverseEc1})$

Choose the option that fills in the blank: If $f(x)$ is a function, then it suffices to know that it __________________________ in order to know it has a (true) inverse.

is one to onehas no relative extremahas no absolute extremais continuous with no relative extremapasses the vertical line test

Any change in the domain of $f(x)$ corresponds to a change in what for $f^{-1}(x)$ ?

Change in range.Whether or not $f^{-1}(x)$ exists.Change in domain.Forces $f^{-1}(x)$ to be discontinuous.Change in the definition (but not context) for $f^{-1}(x)$ .

$(\sage {p1inverseHxfalse},\sage {p1inverseHy})$ $(\sage {p1inverseHy},\sage {p1inverseHx})$ $(\sage {p1inverseHx},\sage {p1inverseHy})$ $(\sage {p1inverseHxfalse},\sage {p1inverseHyfalse})$ $(\sage {p1inverseHyfalse},\sage {p1inverseHxfalse})$

Let $f(x)$ be a relation. If $f^{-1}(x)$ is a function, must it be true that $f(x)$ is a function?

No, if the inverse is a function, then $f(x)$ cannot be a function.Yes, because the inverse can only be defined if $f(x)$ is a function.No, it’s possible the inverse is a function but the original would not be.Yes, $f^{-1}(x)$ being a function means it passes the horizontal line test, so $f(x)$ is therefore a function.Yes, $f^{-1}(x)$ being a function means it passes the vertical line test, so $f(x)$ is therefore a function.

If a function has a relative maximum and a relative minimum, must it have an absolute extrema?

Yes because there must be an absolute extrema between the relative minumum or maximum.No, it’s possible to have relative maximums and minimums, but no absolute extrema.There is not enough information to know.Yes because there must be an absolute extrema beyond the relative minumum/maximum.Yes, because one of the relative extrema must also be an absolute extrema.

If a continuous function has no absolute extrema, and it has a relative maximum, must it have a relative minimum?

Yes.No, it’s possible to have no absolute maximum or minimums, and only relative maximums.No, it’s not possible to have a function with no absolute maximum or minimums, yet still have a relative maximum.No, but if it had a relative minimum, then it would have to have a relative maximum.There is not enough information to answer this question.

If the function $f(x)$ has $\sage {p1POIHc1}$ zeros, then how many zeros must $f(x) + \sage {p1POIHc2}$ have?

$\sage {p1POIHansF}$ $0$ $\sage {p1POIHc1}$ There is not enough information to tell. $\sage {-p1POIHc1}$

If you double the height of $f(x)$ and then move it up by 4 is it the same as doubling the height and then moving it to the left by 4?

There is not enough information to answer.This is never true.This is always true.Sometimes this is true, but definitely not always.

Which of the following represents the translation that moves the graph of $f(x)$ to the left by $\sage {p1TransEc1}$ ?

$f(x) + \sage {p1TransEc1}$ $f(x - \sage {p1TransEc1})$ $f(x + \sage {p1TransEc1})$ $f(x) - \sage {p1TransEc1}$ $f(\sage {p1TransEc1}x)$

Which of the following represents the translation that moves the graph of $f(x)$ up by $\sage {p1TransEc1}$ ?

$f(x) - \sage {p1TransEc1}$ $f(\sage {p1TransEc1}x)$ $f(x - \sage {p1TransEc1})$ $f(x) + \sage {p1TransEc1}$ $f(x + \sage {p1TransEc1})$

Which of the following represents a rigid translation?

$\sage {abs(p3TransEc2)}f(\sage {x-p3TransEc1})$ $f(\sage {p3TransEc1*x}) - \sage {abs(p3TransEc2)}$ $f(\sage {x-p3TransEc1}) + \sage {abs(p3TransEc2)}$ $f(\sage {p3TransEc1*x})$ $\sage {abs(p3TransEc1)}f(\sage {p3TransEc2*x})$

If a function has a single absolute extrema, must it have a relative maximum or a relative minimum?

Yes, the absolute extrema must be either a relative maximum or relative minimum.There is not enough information to know.No because it’s impossible to have a function that has only one absolute extrema.Yes, because all relative extrema must also be absolute extrema.No, it is possible to have an absolute extrema without any relative maximum or minimums.

Is it possible to have a discontinuous function without any relative extrema?

Yes, but only if it is an infinite discontinuity.No, regardless of the type of discontinuity.Yes, but only if it is an infinite or jump discontinuity.Yes, but only if it is a jump discontinuityYes, regardless of the type of discontinuity.

If the function $f(x)$ has $\sage {p1POIMc1}$ zeros, then how many zeros must $f(\sage {x - p1POIMc2})$ have? (Hint, consider this in terms of translations or transformations)

$\sage {abs(p1POIMc2)}$ $\sage {p1POIMc1}$ $\sage {p1POIMansF}$ There is not enough information to tell. $0$

Which of the following accurately describes the sequence of transformations applied to $f(x)$ to obtain $\sage {p1TransM1udmult}f(\sage {p1TransM1lrmult}(\sage {x+p1TransM1rl})) + (\sage {p1TransM1ud})$ ?

The function $f$ $\sage {p1TransM1lrfliptxt}$ horizontally stretched to $\frac {1}{\sage {p1TransM1c3}}$ its original width, shifted $\sage {p1TransM1lrtxt}$ by $\sage {p1TransM1c2}$ , vertically stretched to $\sage {p1TransM1c1}$ times it’s original height, $\sage {p1TransM1udfalsetxt}$ and shifted $\sage {p1TransM1udtxt}$ by $\sage {p1TransM1c4}$ .The function $f$ $\sage {p1TransM1lrfalsetxt}$ horizontally stretched to $\frac {1}{\sage {p1TransM1c1}}$ its original width, shifted $\sage {p1TransM1lrtxt}$ by $\sage {p1TransM1c4}$ , vertically stretched to $\sage {p1TransM1c3}$ times it’s original height, $\sage {p1TransM1udfliptxt}$ and shifted $\sage {p1TransM1udtxt}$ by $\sage {p1TransM1c2}$ .The function $f$ $\sage {p1TransM1lrfliptxt}$ horizontally stretched to $\frac {1}{\sage {p1TransM1c3}}$ its original width, shifted $\sage {p1TransM1lrtxt}$ by $\sage {p1TransM1c2}$ , vertically stretched to $\sage {p1TransM1c1}$ times it’s original height, $\sage {p1TransM1udfliptxt}$ and shifted $\sage {p1TransM1udtxt}$ by $\sage {p1TransM1c4}$ .The function $f$ $\sage {p1TransM1lrfliptxt}$ horizontally stretched to $\frac {1}{\sage {p1TransM1c1}}$ its original width, shifted $\sage {p1TransM1lrtxt}$ by $\sage {p1TransM1c4}$ , vertically stretched to $\sage {p1TransM1c3}$ times it’s original height, $\sage {p1TransM1udfliptxt}$ and shifted $\sage {p1TransM1udtxt}$ by $\sage {p1TransM1c2}$ .The function $f$ $\sage {p1TransM1lrfalsetxt}$ horizontally stretched to $\frac {1}{\sage {p1TransM1c3}}$ its original width, shifted $\sage {p1TransM1lrtxt}$ by $\sage {p1TransM1c2}$ , vertically stretched to $\sage {p1TransM1c1}$ times it’s original height, $\sage {p1TransM1udfliptxt}$ and shifted $\sage {p1TransM1udtxt}$ by $\sage {p1TransM1c4}$ .

Which equation accurately reflects the following sequence of transformations (in the correct order)? The function $g$ is $\sage {p2TransMlrfliptxt}$ horizontally stretched to $\frac {1}{\sage {p2TransMc3}}$ its original width, shifted $\sage {p2TransMlrtxt}$ by $\sage {p2TransMc2}$ , vertically stretched to $\sage {p2TransMc1}$ times it’s original height, $\sage {p2TransMudfliptxt}$ and shifted $\sage {p2TransMudtxt}$ by $\sage {p2TransMc4}$ .

$\sage {p2TransMudmult}g(\sage {p2TransMfalse3inner}) + (\sage {p2TransMud})$ $\sage {p2TransMudmult}g(\sage {p2TransMfalse2inner}) + (\sage {p2TransMud})$ $\sage {p2TransMudmult}g(\sage {p2TransMfalse1inner}) + (\sage {p2TransMud})$ $\sage {p2TransMudmult}g(\sage {p2TransMfalse1inner}) - (\sage {p2TransMud})$ $\sage {p2TransMudmult}g(\sage {p2TransMlrmult}(\sage {x+p2TransMrl})) + (\sage {p2TransMud})$

Suppose $h(x)$ is the result after some other function $f(x)$ has been through the following changes; $\sage {p3TransMlrfliptxt}$ horizontally stretched to $\frac {1}{\sage {p3TransMc3}}$ its original width, shifted $\sage {p3TransMlrtxt}$ by $\sage {p3TransMc2}$ , vertically stretched to $\sage {p3TransMc1}$ times it’s original height, $\sage {p3TransMudfliptxt}$ and shifted $\sage {p3TransMudtxt}$ by $\sage {p3TransMc4}$ . If the point $(\sage {p3TransMc5},\sage {p3TransMc6})$ was on the graph of $f(x)$ , what is the corresponding point on the graph of $h(x)$ ?

It is impossible to determine. $(\sage {p3TransMansfalsex}, \sage {p3TransMansy})$ $(\sage {p3TransMansx}, \sage {p3TransMansfalsey})$ $(\sage {p3TransMansfalsex}, \sage {p3TransMansfalsey})$ $(\sage {p3TransMansx}, \sage {p3TransMansy})$

The function $f(x)$ is transformed and the points $(\sage {p1TransH1x1},\sage {p1TransH1y1})$ and $(\sage {p1TransH1x2},\sage {p1TransH1y2})$ on the graph of $f$ are sent to the points $(\sage {p1TransH1x3},\sage {p1TransH1y3})$ and $(\sage {p1TransH1x4},\sage {p1TransH1y4})$ respectively. Which of following expressions could describe the transformations applied to $f(x)$ ?

$(\sage {p1TransH1c2})f(\sage {p1TransH1f3}) + (\sage {p1TransH1c4})$ $(\sage {p1TransH1c4})f(\sage {p1TransH1f1}) + (\sage {p1TransH1c2})$ $(\sage {p1TransH1c2})f(\sage {p1TransH1f1}) + (\sage {p1TransH1c4})$ $(\sage {p1TransH1c2})f(\sage {p1TransH1f1}) + (\sage {p1TransH1c2*p1TransH1c4})$ $(\sage {p1TransH1c2})f(\sage {p1TransH1f2}) + (\sage {p1TransH1c4})$

Suppose $f(x) = \sage {p1compEfunc1}$ and $g(x) = \sage {p1compEfunc2}$ ; compute $(f \circ g)(\sage {p1compEc1})$ .

$\sage {p1compEans3}$ $\sage {p1compEans4}$ $\sage {p1compEans5}$ $\sage {p1compEans2}$ $\sage {p1compEans1}$

Suppose $f(x) = \sage {p2compEfunc1}$ and $g(x) = \sage {p2compEfunc2}$ ; compute $(f \circ g)(\sage {p2compEc1})$ .

$\sage {p2compEans5}$ $\sage {p2compEans2}$ $\sage {p2compEans4}$ $\sage {p2compEans1}$ $\sage {p2compEans3}$

Which of the following definitions of $f(x)$ and $g(x)$ could result in the function $\sage {p3compEf3}$

$f(x) = \sage {p3compEf1}$ and $g(x) = \sage {p3compEf2false}$ $f(x) = \sage {p3compEf1}$ and $g(x) = \sage {p3compEf2}$ $f(x) = \sage {p3compEf1false}$ and $g(x) = \sage {p3compEf2false}$ $f(x) = \sage {p3compEf1false}$ and $g(x) = \sage {p3compEf2}$ $f(x) = \sage {p3compEf2}$ and $g(x) = \sage {p3compEf1}$

Suppose $f(x) = \sage {compHfunc1}$ , $g(x) = \sage {compHfunc2}$ , and $h(x) = \sage {compHfunc3}$ ; compute $(f \circ g)(\sage {compHc1})\cdot h(x)$ .

$\sage {compHans3}$ $\sage {compHans2}$ $\sage {compHans1}$ $\sage {compHans4}$ $\sage {compHans5}$

You are reviewing a model your company has to calculate the cost to modify one of your software packages for individual usage. You need to update the model due to inflation and changes in the industry, and you have developed a transform that does exactly this. In particular, the updated cost is calculated by $U(x) = \sage {p1compMc1}C(x) - \sage {p1compMc2}$ , where $C(x)$ is the original cost. If a customer had originally been quoted a cost of $ $\sage {p1compMc4}$ for modifying $\sage {p1compMc3}$ software packages, what would the updated cost be?

$ $\sage {p1compMans2}$ $ $\sage {p1compMans5}$ $ $\sage {p1compMans3}$ $ $\sage {p1compMans4}$ $ $\sage {p1compMans1}$

Suppose $f(x)$ has a zero at an $x$ -value of $\sage {p2compMc1}$ .
What would a zero of $g(x)$ be if $g(x) = \sage {p2compMc2}f(\sage {p2compMf1})$ ?

$\sage {p2compMans5}$ $\sage {p2compMans2}$ $\sage {p2compMans3}$ $\sage {p2compMans4}$ $\sage {p2compMans1}$

If $f(x) = \sage {p3compMf1}$ and $g(x) = \sage {p3compMf2}$ , what is $(g\circ f)(x)$ ?

$\sage {p3compMans4}$ $\sage {p3compMans3}$ $\sage {p3compMans2}$ $\sage {p3compMans5}$ $\sage {p3compMans1}$

If $f(x) = \sage {p3compMf1}$ and $g(x) = \sage {p3compMf2}$ , what is $\big ((\sage {p4compMc1})\cdot f + (\sage {p4compMc2})\cdot g\big )(x)?$

$\sage {p4compMans5}$ $\sage {p4compMans1}$ $\sage {p4compMans4}$ $\sage {p4compMans2}$ $\sage {p4compMans3}$

Let $f:\mathbb {R}\rightarrow \mathbb {R}^+$ and $g:(-\infty ,0]\rightarrow \mathbb {R}$ be defined by: $f(x) = x^2+1337$ and $g(x) = \sqrt {(-x)^2}$ .

What is $(g\circ f)(x)$ ?

The composition fails to exist, so $(g\circ f)(x)$ is undefined. $(\sqrt {-x^2})^2+1337)$ $|-x^2 + 1337|$ $\sqrt {-(x^2+1337)^2}$ $|x^2|+ 1337$

A university parking lot designates spaces based on parking permit colors. The lot has 6 rows of 12 parking spaces each. The rows are labelled A-F and the spaces are numbered 1-12 in each row. The different color permits are allowed to park as follows:

Red permits may park in any space.
Orange permits may park in any space in rows B-E.
Green permits are allowed to park in row E spots 6-12 and (any spot) in row F.
Blue permits may only park in row D.

Is the relationship that determines your potential parking space based on your permit color a function?

No.Yes.

What is the parent function of $g(x) = \sage {p1funcEf2} + \sage {p1funcEf1} + \sage {p1funcEf3} + \sage {p1funcEf4}$ ?

$f(x) = \sage {p1funcEans4}$ $f(x) = \sage {p1funcEans2}$ $f(x) = \sage {p1funcEans1}$ $f(x) = \sage {p1funcEans5}$ $f(x) = \sage {p1funcEans3}$

Grades in a particular class are determined using the following numeric scale.


Grade	Point Range	Grade	Point Range	Grade	Point Range

A	555-600	B-	480-499	D+	400-419
A-	540-554	C+	465-479	D	360-399
B+	525-539	C	435-464	D-	340-359
B	500-524	C-	420-434	E	0-339

Clearly, given any point value, a student can determine their letter grade in the class. This means we have a relationship from point values to letter grades.

What is the domain and codomain of this relationship?

Domain: Scores on assignments. Codomain: Letter Grades.Domain: Point Values. Codomain: Letter Grades.Domain: Real numbers. Codomain: Letters.Domain: Letter Grades. Codomain: Scores on assignments.Domain: Letter Grades. Codomain: Point Values.

Consider the relationship that takes in your individual assignment grades for this course, and returns the letter grade you earn at the end of the semester. Is this relation a function?

No, there is one point value for each letter grade.There is not enough information here to tell.Yes, there is one point value for each letter grade.No, there is only one possible letter grade for each point value.Yes, there is only one possible letter grade for each point value.

Describe the following set with its ”English-translation”:

$\{x:x\in \mathbb {R}, 17\leq x< 38\}$

The set of numbers between 17 and 38.The set of all rational numbers between 17 and 38, including 17.The set of $x$ such that $x$ is between 17 and 38.The set of real numbers $x$ such that $17 \leq x \leq 38$ The set of $x$ such that $x$ is a real number, and $x$ is strictly less than 38, but no smaller than 17.

When someone calls you and they are in your contacts, your phone uses their phone number to display their name. What are the domain and codomain of this relationship?

Domain: names in your contacts; Codomain: phone numbersDomain: phone numbers in your contacts; Codomain: namesDomain: phone numbers; Codomain: namesDomain: names; Codomain: phone numbers

Which of the following is best described as a mathematical expression?

force is mass times acceleration $\sage {p2funcMf2} = \sage {p2funcMf2c}$ $\sqrt {a^2 + b^2}$ $y = \sage {p2funcMf1}$ $\sage {p2funcMf3} \geq y$

Which coordinates describe a point that is $\sage {p1graphEc2}$ units $\sage {p1graphEd2}$ and $\sage {p1graphEc1}$ units to the $\sage {p1graphEd1}$ of the origin?

$\sage {p1graphEans4}$ $\sage {p1graphEans2}$ $\sage {p1graphEans3}$ $\sage {p1graphEans5}$ $\sage {p1graphEans1}$

Simplify the following set: $\{x\in \mathbb {R}^+\cup \{0\} : x \leq 0 \}$

$x \leq 0$ $(0, \infty )$ $(-\infty , \infty )$ $(-\infty , 0)$ $\{ 0 \}$

Which of the following represents the set of all rational numbers that are no larger (more positive) than $\sage {p1funcHc1}$ ? (Note: The set of rational numbers is denoted $\mathbb {Q}$ )

$\{ x\in \mathbb {R}: x \leq \sage {p1funcHc1} \}$ $\{ x\in \mathbb {Q}: x < \sage {p1funcHc1} \}$ $\{ x\in \mathbb {Q}: x \leq \sage {p1funcHc1} \}$ $(-\infty , \sage {p1funcHc1}]$ $(-\infty , \sage {p1funcHc1})$

Fill in the blank: Given a graph of a function, you can determine the _________________ of points of interest.

exact coordinates; if they are close enough to the axes importancecoordinates (within an error of $\pm$ 1 unit)meaningexistance

Which of the following has an absolute maximum?

Which of the following graphs would most properly be said to have the parent function $f(x) = e^x$ ?

Given the following graph:

What are the approximate (x,y) coordinates of the local minimum(s), if any exist? (Select all that are local minimums; keep in mind we are asking for an approximation, not precise values)

$\left (-\frac {9}{4},-\frac {5}{2}\right )$ and $\left ( \frac {1}{2}, \frac {1}{4} \right )$ $\left ( \frac {1}{2}, \frac {1}{4} \right )$ $\left ( -\frac {3}{4}, \frac {19}{10} \right )$ $\left (-\frac {9}{4},-\frac {5}{2}\right )$ There are no local minimums.

Given the following graph:

What are the approximate (x,y) coordinates of the absolute maximum(s), if any exist?

$\left (-\frac {9}{4},-\frac {5}{2}\right )$ $\left ( \frac {1}{2}, \frac {1}{4} \right )$ $\left ( -\frac {3}{4}, \frac {19}{10} \right )$ $\left (\frac {5}{4},6\right )$ There are no absolute maximums.

Given the following graph:

On what interval(s) is the function f(x) decreasing? (Approximate the endpoints if needed.)

$\left (-\infty , -\frac {9}{4} \right ) \cup \left (-\frac {3}{4}, \frac {1}{2} \right )$ $\left (-\frac {9}{4}, -\frac {3}{4} \right )$ $\left (-\frac {3}{4}, \frac {1}{2} \right )$ $\left (-\infty , -\frac {9}{4} \right )$ $\left (\frac {5}{4}, \infty \right )$

Consider the following graph:

Does the graph have any relative extrema? If so, (approximately) where?

Yes; a minimum at $(-3,0)$ Yes; a minimum at $(-3,0)$ and a maximum at $(-2,3)$ Yes; a minimum at $(-3,0)$ and a maximum at $(-2,1)$ Yes; a maximum at $(-2,3)$ Yes; a maximum at $(-2,1)$

Consider the following graph:

Does the graph have any discontinuities? If so, what type?

Yes, it has both infinite and jump discontinuities. Yes, it (only) has jump discontinuities. Yes, it (only) has infinite discontinuities. Yes, it has jump, infinite, and hole discontinuities. Yes, it has infinite and hole discontinuities.

Consider the following graph:

Does this graph have any absolute extrema?

No, because the graph extendws infinitely up and down. No, because the graph extendws infinitely left and right. No, because the graph has discontinuities. Yes, it has an absolute minimum. Yes, it has an absolute minimum and maximum.

Which of the following graphs depicts a continuous relationship?

Identify the coordinates of the points shown on the graph.

a: (0,4); b: (2,1); c: (-5, -3) a: (-4,0); b: (-1,2); c: (-3, 5) a: (0,-4); b: (-2,-1); c: (5, 3) a: (4,0); b: (1,2); c: (-3,-5) a: (0,4); b: (-2,1); c: (5, -3)

For the following graph which is the correct parent function?

$f(x)=x^3$ $f(x)=x^2$ $f(x) = |x|$ $f(x) = e^x$ $f(x) = \sqrt [3]{x}$