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Determine whether or not the mapping diagram represents a function. Explain your reasoning. If the mapping does
represent a function, state the domain, range, and represent the function as a set of ordered pairs.
The mapping \(M\) is not a function since ‘Tennant’ is matched with both ‘Eleven’ and ‘Twelve.’
Determine whether or not the mapping diagram represents a function. Explain your reasoning. If the mapping does
represent a function, state the domain, range, and represent the function as a set of ordered pairs.
The mapping \(C\) is a function since each input is matched with only one output. The domain of \(C\) is \(\{ \text {Hartnell}, \text {Cushing}, \text {Hurndall}, \text {Troughton} \}\) and the range is \(\{\text {One}, \text {Two} \}\). We can
represent \(C\) as the following set of ordered pairs: \(\{ (\text {Hartnell}, \text {One}), (\text {Cushing}, \text {One}), (\text {Hurndall}, \text {One}), (\text {Troughton}, \text {Two}) \}\)
In Exercises tablefirst - tablelast, determine whether or not the data in the given table represents \(y\) as a function of \(x\). Explain your reasoning. If
the mapping does represent a function, state the domain, range, and represent the function as a set of ordered
pairs.
In this case, \(y\) is not a function of \(x\) since there are \(x\) values matched with more than one \(y\) value. For instance, \(1\) is matched both
to \(1\) and \(-1\).
Suppose \(W\) is the set of words in the English language and we set up a mapping from \(W\) into the set of natural numbers \(\mathbb {N}\) as
follows: \(\text {word} \rightarrow \text {number of letters in the word}\). Explain why this mapping is a function. What would you need to know to determine the range of the
function?
The mapping is a function since given any word, there is only one answer to ‘how many letters are in the word?’ For the range,
we would need to know what the length of the longest word is and whether or not we could find words of all the lengths
between \(1\) (the length of the word ‘a’) and it. See here.
Suppose \(L\) is the set of last names of all the people who have served or are currently serving as the President of the United
States. Consider the mapping from \(L\) into \(\mathbb {N}\) as follows: \(\text {last name} \rightarrow \text {number of their presidency}\). For example, \(\text {Washington} \rightarrow 1\) and \(\text {Obama} \rightarrow 44\). Is this mapping a function? What if
we use full names instead of just last names?
Research what Grover Cleveland and Donald Trump have in
common.
Since Grover Cleveland was both the 22nd and 24th POTUS, neither mapping described in this exercise is a function. Donald
Trump would also have two “outputs”.
Under what conditions would the time of day be a function of the outdoor temperature?
The outdoor temperature could never be the same for more than two different times - so, for example, it could always be
getting warmer or it could always be getting colder.
For the functions \(f\) described in Exercises buildfunctionfirst - buildfunctionlast, find \(f(2)\), and find and simplify an expression for \(f(x)\) that takes a real number \(x\) and
performs the following three steps in the order given.
In Exercises equfunctionfirst - equfunctionlast, determine whether or not the following equation represents \(y\) as a function of \(x\).
\(y = x^{3} - x\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
\(y = \sqrt {x - 2}\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
\(x^{3}y = -4\)
Yes, \(y\) is a function of \(x\).
Yes, in fact, \(y=\frac {-4}{x^3}\).
No, \(y\) is a not a function of \(x\).
\(x^{2} - y^{2} = 1\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
We cannot solve for y without using \(\pm \) when we take the square root of both
sides of our equation.
\(y = \frac {x}{x^{2} - 9}\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
\(x = -6\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
\(x = y^2 + 4\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
We cannot solve for y without using \(\pm \) when we take the square root of
both sides of our equation.
\(y = x^2 + 4\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
\(x^2 + y^2 = 4\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
We cannot solve for y without using \(\pm \) when we take the square root of
both sides of our equation.
\(y = \sqrt {4-x^2}\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
\(x^2 - y^2 = 4\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
We cannot solve for y without using \(\pm \) when we take the square root of
both sides of our equation.
\(x^3 + y^3 = 4\)
Yes, \(y\) is a function of \(x\).
Yes, in fact, \(y=\sqrt [3]{4-x^3}\).
No, \(y\) is a not a function of \(x\).
\(2x + 3y = 4\)
Yes, \(y\) is a function of \(x\).
Yes, in fact, \(y=-\frac {2}{3}x+\frac {4}{3}\).
No, \(y\) is a not a function of \(x\).
\(2xy = 4\)
Yes, \(y\) is a function of \(x\).
Yes, in fact, \(y=\frac {4}{2x}\).
No, \(y\) is a not a function of \(x\).
\(x^2 = y^2\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
We cannot solve for y without using \(\pm \) when we take the square root of
both sides of our equation.
Exercises setfunctionfirst - setfunctionlast give a set of points in the \(xy\)-plane. Determine if \(y\) is a function of \(x\). If so, state the domain and
range.
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
domain \(= \{x \, | \, \text {$x$ is any integer} \}\) range \(= \{y \, | \, \text { $y$ is the square of an integer}\}\)
\(\{ (-2, y) \, | \, -3 < y < 4\}\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
Notice \(-2\) has many“outputs”.
\(\{ (x,3) \, | \, -2 \leq x < 4\}\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
domain \(= \{x \, | \, -2 \leq x < 4 \} = [-2, 4)\), range = {\(3\)}
\(\{ \left (x,x^2\right ) \, | \, \text {$x$ is a real number} \}\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
domain \(= \{x \, | \, \text {$x$ is a real number} \} = (-\infty , \infty )\) range \(= \{y \, | \, y \geq 0 \} = [0,\infty )\)
\(\{ \left (x^2,x\right ) \, | \, \text {$x$ is a real number} \}\)
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
Notice \((4,2)\) and \((4,-2)\) are both in this set, for example.
The Vertical Line Test is a quick way to determine from a graph if the vertical axis variable is a function of the horizontal
axis variable. If we are given a graph and asked to determine if the horizontal axis variable is a function of the vertical axis
variable, we can use horizontal lines instead of vertical lines to check. Using Theorem VLT as a guide, formulate a ‘Horizontal Line
Test.’ (We’ll refer back to this exercise in Section InverseFunctions.)
Horizontal Line Test: A graph on the \(xy\)-plane represents \(x\) as a function of \(y\) if and only if no horizontal line intersects the
graph more than once.
In Exercises graphfunctionfirstxy - graphfunctionlastxy, determine whether or not the graph suggests \(y\) is a function of \(x\). For the ones which do, state the domain and
range.
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
Yes, \(y\) is a function of \(x\).No, \(y\) is a not a function of \(x\).
Notice the vertical line test fails many places.
Determine which of the graphs, if any, in Exercises graphfunctionfirstxy - graphfunctionlastxy, suggest \(x\) is a function of \(y\). (Feel free to use Exercise HLTExercise.) If it does, state
the domain and range.
domain = \((-\infty , \infty )\) and range = \([1, \infty )\)
In Exercises graphfunctionfirstvw - graphfunctionlastvw, determine whether or not the graph suggests \(w\) is a function of \(v\). For the ones which do, state the domain and
range.
Yes, \(w\) is a function of \(v\).No, \(w\) is a not a function of \(v\).
domain = \([2, \infty )\) range = \([0, \infty )\)
Yes, \(w\) is a function of \(v\).No, \(w\) is a not a function of \(v\).
Determine which of the graphs, if any, in Exercises graphfunctionfirstvw - graphfunctionlastvw, suggest \(v\) is a function of \(w\). (Feel free to use Exercise HLTExercise.) If it does, state
the domain and range.
Only number graphfunctionfirstvw represents \(v\) as a function of \(w\); domain = \([0, \infty )\) and range = \([2, \infty )\)
In Exercises graphfunctionfirsttT - graphfunctionlasttT, determine whether or not the graph suggests \(T\) is a function of \(t\). For the ones which do, state the domain and
range.
Yes, \(T\) is a function of \(t\).No, \(T\) is a not a function of \(t\).
domain = \([-2, \infty )\) range = \([-3, \infty )\)
Yes, \(T\) is a function of \(t\).No, \(T\) is a not a function of \(t\).
The vertical line test fails in many places.
Determine whether or not the graph suggests \(T\) is a function of \(t\) If it does, state the domain and range.
Yes, \(T\) is a function of \(t\).No, \(T\) is a not a function of \(t\).
domain = \((-5, 4)\) range = \((-4, 4)\)
Determine whether or not the graph suggests \(T\) is a function of \(t\) If it does, state the domain and range.
Yes, \(T\) is a function of \(t\).No, \(T\) is a not a function of \(t\).
domain = \([0,3) \cup (3,6]\) range = \((-4,-1] \cup [0,4]\)
Determine which of the graphs, if any, in Exercises graphfunctionfirsttT - graphfunctionlasttT, suggest \(t\) is a function of \(T\). (Feel free to use Exercise HLTExercise.) If it does, state
the domain and range.
In Exercises graphfunctionfirstHs - graphfunctionlastHs, determine whether or not the graph suggests \(H\) is a function of \(s\). For the ones which do, state the domain and
range.
Yes, \(H\) is a function of \(s\).No, \(H\) is a not a function of \(s\).
Determine which, if any, of the graphs in numbers graphfunctionfirstHs - graphfunctionlastHs represent \(s\) as a function of \(H\). For the ones which do, state the domain
and range. (Feel free to use Exercise HLTExercise.)
Only number graphfunctionfirstHs3 represents \(s\) as a function of \(H\); domain = \((-\infty , 3]\) and range = \([-2, \infty )\)
In Exercises graphfunctionfirstut - graphfunctionlastut, determine whether or not the graph suggests \(u\) is a function of \(t\). For the ones which do, state the domain and
range.
Yes, \(u\) is a function of \(t\).No, \(u\) is a not a function of \(t\).
Yes, \(u\) is a function of \(t\).No, \(u\) is a not a function of \(t\).
domain = \([-3,3]\) range = \([-2,2]\)
Yes, \(u\) is a function of \(t\).No, \(u\) is a not a function of \(t\).
The vertical line test fails at \(t=2\).
Yes, \(u\) is a function of \(t\).No, \(u\) is a not a function of \(t\).
domain = \((-\infty , \infty )\) range = \(\{2\}\)
Determine which, if any, of the graphs in numbers graphfunctionfirstut - graphfunctionlastut represent \(t\) as a function of \(u\). For the ones which do, state the domain
and range. (Feel free to use Exercise HLTExercise.)
Only number graphfunctionfirstut3 represents \(t\) as a function of \(u\); domain = \((-\infty , \infty )\) and range=\(\{2 \}\).
In Exercises sketchgraphfirst - sketchgraphlast, graph each function by making a table, plotting points, and using a graphing utility (if needed.) Use the
independent variable as the horizontal axis label and the default ‘\(y\)’ label for the vertical axis label. State the domain and range
of each function.
Let \(F = \{ (t, t^2) \, | \, \text {$t$ is a real number} \}\). Find \(F(4)\) and solve \(F(x) = 4\).
Elements of \(F\) are of the form \((x, F(x))\).
\(F(4) = 4^2 = 16\) (when \(t = 4\)), the solutions to \(F(x) = 4\) are \(x = \pm 2\) (when \(t = \pm 2\)).
Let \(G = \{ (2t, t+5) \, | \, \text {$t$ is a real number} \}\). Find \(G(4)\) and solve \(G(x) = 4\).
Elements of \(G\) are of the form \((x, G(x))\).
\(G(4) = 7\) (when \(t = 2\)), the solution to \(G(t) = 4\) is \(x = -2\) (when \(t = -1\))
The area enclosed by a square, in square inches, is a function of the length of one of its sides \(\ell \), when measured in inches.
This function is represented by the formula \(A(\ell ) = \ell ^2\) for \(\ell > 0\). Find \(A(3)\) and solve \(A(\ell ) = 36\). Interpret your answers to each. Why is \(\ell \) restricted to
\(\ell > 0\)?
\(A(3) = 9\), so the area enclosed by a square with a side of length \(3\) inches is \(9\) square inches. The solutions to \(A(\ell ) = 36\) are \(\ell = \pm 6\). Since \(\ell \) is restricted
to \(\ell > 0\), we only keep \(\ell = 6\). This means for the area enclosed by the square to be \(36\) square inches, the length of the side needs to be \(6\)
inches. Since \(\ell \) represents a length, \(\ell > 0\).
The area enclosed by a circle, in square meters, is a function of its radius \(r\), when measured in meters. This
function is represented by the formula \(A(r) = \pi r^2\) for \(r > 0\). Find \(A(2)\) and solve \(A(r) = 16\pi \). Interpret your answers to each. Why is \(r\) restricted to
\(r > 0\)?
\(A(2) = 4\pi \), so the area enclosed by a circle with radius \(2\) meters is \(4\pi \) square meters. The solutions to \(A(r) = 16\pi \) are \(r = \pm 4\). Since \(r\) is restricted to \(r > 0\), we only
keep \(r = 4\). This means for the area enclosed by the circle to be \(16\pi \) square meters, the radius needs to be \(4\) meters. Since \(r\) represents
a radius (length), \(r > 0\).
The volume enclosed by a cube, in cubic centimeters, is a function of the length of one of its sides \(s\), when measured in
centimeters. This function is represented by the formula \(V(s) = s^3\) for \(s > 0\). Find \(V(5)\) and solve \(V(s) = 27\). Interpret your answers to each. Why is \(s\)
restricted to \(s > 0\)?
\(V(5) = 125\), so the volume enclosed by a cube with a side of length \(5\) centimeters is \(125\) cubic centimeters. The solution to \(V(s) = 27\) is \(s = 3\).
This means for the volume enclosed by the cube to be \(27\) cubic centimeters, the length of the side needs to \(3\) centimeters. Since \(x\)
represents a length, \(x > 0\).
The volume enclosed by a sphere, in cubic feet, is a function of the radius of the sphere \(r\), when measured in feet. This
function is represented by the formula \(V(r) =\frac {4\pi }{3} r^{3}\) for \(r > 0\). Find \(V(3)\) and solve \(V(r) = \frac {32\pi }{3}\). Interpret your answers to each. Why is \(r\) restricted to
\(r > 0\)?
\(V(3) = 36\pi \), so the volume enclosed by a sphere with radius \(3\) feet is \(36\pi \) cubic feet. The solution to \(V(r) = \frac {32\pi }{3}\) is \(r = 2\). This means for the
volume enclosed by the sphere to be \(\frac {32\pi }{3}\) cubic feet, the radius needs to \(2\) feet. Since \(r\) represents a radius (length), \(r > 0\).
The height of an object dropped from the roof of an eight story building is modeled by the function: \(h(t) = -16t^2 + 64\), \(0 \leq t \leq 2\). Here, \(h(t)\) is the height of
the object off the ground, in feet, \(t\) seconds after the object is dropped. Find \(h(0)\) and solve \(h(t) = 0\). Interpret your answers to each. Why is
\(t\) restricted to \(0 \leq t \leq 2\)?
\(h(0) = 64\), so at the moment the object is dropped off the building, the object is \(64\) feet off of the ground. The solutions to \(h(t) = 0\) are \(t = \pm 2\). Since we
restrict \(0 \leq t \leq 2\), we only keep \(t = 2\). This means \(2\) seconds after the object is dropped off the building, it is \(0\) feet off the ground. Said
differently, the object hits the ground after \(2\) seconds. The restriction \(0 \leq t \leq 2\) restricts the time to be between the moment the object is
released and the moment it hits the ground.
The temperature in degrees Fahrenheit \(t\) hours after 6 AM is given by \(T(t) = -\frac {1}{2} t^2 + 8t+3\) for \(0 \leq t \leq 12\). Find and interpret \(T(0)\), \(T(6)\) and \(T(12)\).
\(T(0) = 3\), so at 6 AM (\(0\) hours after 6 AM), it is \(3^{\circ }\) Fahrenheit. \(T(6) = 33\), so at noon (\(6\) hours after 6 AM), the temperature is \(33^{\circ }\) Fahrenheit. \(T(12) = 27\), so at 6
PM (\(12\) hours after 6 AM), it is \(27^{\circ }\) Fahrenheit.
The function \(C(x) = x^2-10x+27\) models the cost, in hundreds of dollars, to produce \(x\)thousand pens. Find and interpret \(C(0)\), \(C(2)\) and
\(C(5)\).
\(C(0) = 27\), so to make \(0\) pens, it costs (This is called the ‘fixed’ or ‘start-up’ cost. We’ll revisit this concept in Example PortaBoyCost in Section ConstantandLinearFunctions.)\(\$ 2700\).
\(C(2) = 11\), so to make \(2000\) pens, it costs \(\$1100\). \(C(5) = 2\), so to make \(5000\) pens, it costs \(\$2000\).
Using data from the Bureau of Transportation Statistics, the average fuel economy in miles per gallon for passenger cars in
the US can be modeled by \(E(t) = -0.0076t^2+0.45t + 16\), \(0 \leq t \leq 28\), where \(t\) is the number of years since \(1980\). Use a calculator to find \(E(0)\), \(E(14)\) and \(E(28)\). Round your answers to two
decimal places and interpret your answers to each.
\(E(0) = 16.00\), so in 1980 (\(0\) years after 1980), the average fuel economy of passenger cars in the US was \(16.00\) miles per gallon. \(E(14) = 20.81\), so in
1994 (\(14\) years after 1980), the average fuel economy of passenger cars in the US was \(20.81\) miles per gallon. \(E(28) = 22.64\), so
in 2008 (\(28\) years after 1980), the average fuel economy of passenger cars in the US was \(22.64\) miles per gallon.
The perimeter of a square, in centimeters, is four times the length of one if its sides, also measured in centimeters.
Represent the function \(P\) which takes as its input the length of the side of a square in centimeters, \(s\) and returns the perimeter of
the square\(P(s) = 4s\), \(s > 0\). in inches, \(P(s)\) using a formula.
\(P(s) = 4s\), \(s > 0\).
The circumference of a circle, in feet, is \(\pi \) times the diameter of the circle, also measured in feet. Represent the function \(C\)
which takes as its input the length of the diameter of a circle in feet, \(D\) and returns the circumference of a circle in inches, \(C(D)\) using
a formula.
\(C(D) = \pi D\), \(D > 0\).
Suppose \(A(P)\) gives the amount of money in a retirement account (in dollars) after 30 years as a function of the amount of the
monthly payment (in dollars), \(P\).
What does \(A(50)\) mean?
The amount in the retirement account after 30 years if the monthly payment is \(\$50\).
What is the significance of the solution to the equation \(A(P) = 250000\)?
The solution to \(A(P) = 250000\) is what the monthly payment needs to be in order to have \(\$250,\!000\) in the retirement account after 30 years.
Explain what each of the following expressions mean: \(A(P+50)\), \(A(P)+50\), and \(A(P) + A(50)\).
\(A(P+50)\) is how much is in the retirement account in 30 years if \(\$ 50\) is added to the monthly payment \(P\). \(A(P)+50\) represents the amount of
money in the retirement account after 30 years if \(\$P\) is invested each month plus an additional \(\$50\). \(A(P)+A(50)\) is the sum of money
from two retirement accounts after 30 years: one with monthly payment \(\$P\) and one with monthly payment
\(\$50\).
Suppose \(P(t)\) gives the chance of precipitation (in percent) \(t\) hours after 8 AM.
Write an expression which gives the chance of precipitation at noon.
Since noon is \(4\) hours after 8 AM, \(P(4)\) gives the chance of precipitation at noon.
Write an inequality which determines when the chance of precipitation is more than \(50 \%\).
We would need to solve \(P(t) \geq 50 \%\) or \(P(t) \geq 0.5\).
Explain why the graph in Exercise graphfunctionfirstvw suggests that not only is \(v\) as a function of \(w\) but also \(w\) is a function of \(v\). Suppose \(w = f(v)\) and \(v = g(w)\). That
is, \(f\) is the name of the function which takes \(v\) values as inputs and returns \(w\) values as outputs and \(g\) is the name of the
function which does vice-versa. Find the domain and range of \(g\) and compare these to the domain and range of
\(f\).
The graph in question passes the horizontal line test meaning for each \(w\) there is only one \(v\). The domain of \(g\) is \([0, \infty )\) (which is the
range of \(f\)) and the range of \(g\) is \([2, \infty )\) which is the domain of \(f\).
Sketch the graph of a function with domain \((-\infty , 3) \cup [4,5)\) with range \(\{ 2 \} \cup (5, \infty )\).
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