Previously, we learned how to construct a linear relationship between two variables. Now, we'll explore how to express such relationships using function notation.
For example, suppose the height of a tree, in centimeters, months after planting it is given by the linear equation
According to this linear model,
the tree was high when it was planted, and
the tree grows by each month.
In practice, we tend to model linear relationships of this kind using function notation, as follows:
This gives us the following additional information:
The height of the tree depends on the time since the tree was planted. For this reason, we call the dependent variable, and the independent variable.
Writing makes it clear that the height of the tree depends on the time and not vice versa.
We can use our function to make predictions about the tree's height. For example, to find the height of the tree after months, we substitute into the height function:
Thus, the tree will be centimeters tall after months.
The temperature of a liquid, in degrees Fahrenheit, minutes after it starts cooling is given by
What will the temperature be after minutes?
We are told that the temperature is given by the function
where represents the number of minutes after cooling starts.
We substitute into the temperature function:
Thus, the temperature will be after minutes.
The amount $F$ of fertilizer in storage, in kilograms, $t$ days after a restocking is given by
\[ F(t) = 3000 + 150t. \] How much fertilizer will be in storage after $10$ days?
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a
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$4480$ |
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b
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$4520$ |
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c
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$4450$ |
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d
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$4550$ |
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e
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$4500$ |
The temperature $T$ of a chemical solution, in degrees Celsius, $t$ minutes after the reaction starts is given by
\[ T(t) = 20 - 2.5t. \] What will the temperature be after $12$ minutes?
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a
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$-12$ |
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b
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$-11$ |
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c
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$-10$ |
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d
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$-9$ |
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e
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$-8$ |
Suppose that the daily profit in dollars, of a particular restaurant can be represented by the function
where is the number of guests visiting the restaurant that day.
Now, suppose we want to interpret the following statement in this context:
To interpret this statement, we begin by breaking it down:
the input represents the number of visitors, while
the output is the corresponding profit.
Therefore, the statement can be interpreted as follows:
When visitors come to the restaurant, the restaurant earns a profit.
The number of people living in Pasadena, California, can be modeled by the function where is the number of years since the year What does the statement mean in this context?
First, we interpret and separately:
represents the number of people living in Pasadena years after that is, in
represents the number of people living in Pasadena years after that is, in
Therefore, the statement means the following:
"More people lived in Pasadena in compared to "
MJ takes a free throw during a basketball game. The function $h(d) $ models the ball's height from the ground, in meters, when the distance between MJ and the ball is $d$ meters. What does the expression $h(y) =5$ mean in this context?
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a
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The ball reaches a height of $5$ meters after $y$ seconds |
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b
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The ball travels a total distance of $5$ meters |
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c
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The ball was $5$ meters from the ground when the distance between MJ and the ball was $y$ meters |
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d
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The ball was $y$ meters from the ground when the distance between MJ and the ball was $5$ meters |
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e
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The ball reaches a height of $y$ meters after $5$ seconds |
The function $W(t)$ models the amount of water in a tank, in gallons, after $t$ hours of consumption. What does the statement $W(12) < W(3)$ mean in this context?
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a
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The tank contained less water after $3$ hours than after $12$ hours of consumption. |
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b
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The tank contained more water after $3$ hours than after $12$ hours of consumption. |
|
c
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There were more than $12$ gallons of water in the tank after $3$ hours of consumption. |
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d
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There were exactly $12$ gallons of water in the tank after $3$ hours of consumption. |
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e
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There were fewer than $3$ gallons of water in the tank after $12$ hours of consumption. |
The function $w(t)$ models the weight of Amy's dog, in pounds, when he was $t$ months old. What does the expression $w(10) =w(14)-2$ mean in this context?
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a
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The weight of Amy's dog was $10+2$ pounds at the age of $14$ months. |
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b
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Amy's dog lost $2$ pounds in weight between the ages of $10$ and $14$ months. |
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c
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Amy's dog gained $14$ pounds in weight between the ages of $2$ and $10$ months. |
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d
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The weight of Amy's dog was $14-2$ pounds at the age of $10$ months. |
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e
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Amy's dog gained $2$ pounds in weight between the ages of $10$ and $14$ months. |
Recall that the daily profit of a restaurant, in dollars, can be represented by the function
where is the number of visitors to the restaurant that day.
We can interpret the various parts of the function as follows:
The constant is the amount of profit the restaurant earns when visitors come to the restaurant. In other words, the restaurant loses if no visitors come.
The coefficient is the amount of additional profit that the restaurant earns for each visitor that comes to the restaurant.
The number of hours per year that people in a particular country spent on social media can be modeled by the function where is the number of years since the year What does the constant mean in this context?
Notice that by substituting into our function, we get
The value corresponds to the year So in the people in this country spent hours per year on social media.
Therefore, the constant term represents the number of hours per year that people in this country spent on social media in
The function $h(x) = 30x+57$ gives the average number of hours per year that people in the USA historically have spent on the Internet, where $x$ is the number of years since the year $2000.$ What does the constant $57$ mean in this context?
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a
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The amount of time spent on the internet decreases by $57$ hours every year. |
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b
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In $2000,$ people spent an average of $57$ hours per year using the Internet. |
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c
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The amount of time spent on the internet increases by $57$ hours every year. |
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d
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People spent an average of $57$ hours per year using the Internet. |
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e
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In $2001,$ people spent an average of $57$ hours per year using the Internet. |
The fare, in dollars, that a taxi company charges customers for an $x$-kilometer ride is modeled by the function $f(x) = 0.5 x +5.$ Which of the following statements are true?
- The fare for a $1\, \mathrm{km}$ ride is $\$5$
- The fare for a $10\, \mathrm{km}$ ride is $\$10$
- A customer can get a $4 \, \mathrm{km}$ ride by paying $\$7.$
|
a
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III only |
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b
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I only |
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c
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I and III only |
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d
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I and II only |
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e
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II and III only |
The linear function $f(x)$ gives the temperature in a certain city in degrees Celsius, where $x$ is the number of weeks after the summer season ends. On the day the summer season ended, the temperature was $42^\circ.$ Additionally, $2$ weeks after the summer season ended, the temperature decreased to $28^\circ.$ Find $f(x).$
|
a
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$f(x) = 42-7x$ |
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b
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$f(x) = 28+x$ |
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c
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$f(x) = 35-x$ |
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d
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$f(x) = 42+14x$ |
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e
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$f(x) = 42-28x$ |
Alan saves a certain amount of money each month. The total savings he has can be modeled by the function where is the number of months since he started saving. How can the following statement be written using the function notation?
"After months, Alan saved more than he saved after months."
is the amount of money that Alan saved after months. So,
represents the amount of money that Alan saved after months, and
represents the amount of money that Alan saved after months.
After months, Alan saved more than he did after months, so is more than Therefore, we have
Ken's annual salary can be modeled by the function $S(x)$ where $x$ is the number of years Ken has worked. How can the following statement be written using the function notation?
"After $4$ years, Ken's salary was $\$10\,000$ more than after $2$ years."
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a
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$S(2) = S(4) + 10\,000$ |
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b
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$S(2) > S(4) + 10\,000$ |
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c
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$S(4) = S(2) + 10\,000$ |
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d
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$S(4) > S(2) + 10\,000$ |
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e
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$S(2) + S(4) = 10\,000$ |
Lisa saves a certain amount of money each week. The amount she saves can be modeled by the function $M(x),$ where $x$ is the number of weeks since she started saving. How can the following statement be written using the function notation?
"After $10$ weeks, Lisa saved more than $\$240.$"
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a
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$M(240) = 10$ |
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b
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$M(10) > 240$ |
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c
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$M(240) < 10$ |
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d
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$M(240) > 10$ |
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e
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$M(10) < 240$ |