Functions and function notation - HiSET
Card 1 of 68
is a linear function defined on the set of real numbers. Four values of 
are given in the following table:
Give the definition for 
.
Give the definition for
Tap to reveal answer
By the two-point formula, the equation of a line through points 
is
,
where 
, the slope of the line. The choice of the two points from the given four is arbitrary, so we will select the middle two ordered pairs. Substituting 
, then
The first formula becomes
Since 
, solve for 
. First, distribute 
throughout the difference at right:
Isolate 
by adding 10 to both sides:
Replacing, the definition of 
is
.
By the two-point formula, the equation of a line through points
where 
The first formula becomes
Since
Isolate
Replacing, the definition of
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The daily pay in U.S. Dollars for a certain job is defined as the following function 
, where 
equals time in hours:
If an employee works for 7 hours in a day, how much is he or she paid?
The daily pay in U.S. Dollars for a certain job is defined as the following function 
If an employee works for 7 hours in a day, how much is he or she paid?
Tap to reveal answer
The above function can be understood as, "An employee is paid $15 for each hour he or she works, plus a flat amount of $10."
If an employee works 7 hours, we can find the amount that he or she is paid by plugging in 7 for "Hours" in the equation:
Adhering to order of operations, we next find the product of 15 and 7:
Finally, we find the sum of 105 and 10:
The employee is paid $115.00 for seven hours' work.
The above function can be understood as, "An employee is paid $15 for each hour he or she works, plus a flat amount of $10."
If an employee works 7 hours, we can find the amount that he or she is paid by plugging in 7 for "Hours" in the equation:
Adhering to order of operations, we next find the product of 15 and 7:
Finally, we find the sum of 105 and 10:
The employee is paid $115.00 for seven hours' work.
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Consider the scenario below:
Helen is a painter. It takes her 3 days to make each painting. She has already made 6 paintings. Which of the following functions best models the number of paintings she will have after 
days?
Consider the scenario below:
Helen is a painter. It takes her 3 days to make each painting. She has already made 6 paintings. Which of the following functions best models the number of paintings she will have after
Tap to reveal answer
The question asks, "Which of the following functions best models the number of paintings she will have after 
days?"
From this, you know that the variable 
represents the number of days, and that 
represents the number of paintings she makes as a function of days spent working.
If it takes 3 days to make a painting, each day results in 
paintings. Therefore, we have a linear relationship with slope 
.
Additionally, she begins with 6 paintings. Therefore, even when zero days are spent working on paintings, she will have 6 paintings. In other words, 
. This means the y-intercept is 6.
As a result, the function will be
which can be rewritten as
The question asks, "Which of the following functions best models the number of paintings she will have after
From this, you know that the variable 
If it takes 3 days to make a painting, each day results in
Additionally, she begins with 6 paintings. Therefore, even when zero days are spent working on paintings, she will have 6 paintings. In other words,
As a result, the function will be
which can be rewritten as
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is a linear function defined on the set of real numbers. Four values of are given in the following table:
Give the definition for .
Give the definition for
Tap to reveal answer
By the two-point formula, the equation of a line through points is
,
where , the slope of the line. The choice of the two points from the given four is arbitrary, so we will select the middle two ordered pairs. Substituting , then
The first formula becomes
Since , solve for . First, distribute throughout the difference at right:
Isolate by adding 10 to both sides:
Replacing, the definition of is
.
By the two-point formula, the equation of a line through points
where
The first formula becomes
Since
Isolate
Replacing, the definition of
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The plot shows the graphs of five different equations. Using shape and location, determine which graphed line corresponds to the equation 
.
The plot shows the graphs of five different equations. Using shape and location, determine which graphed line corresponds to the equation 
Tap to reveal answer
The power of 
and 
are both 1, so this is a linear equation of order 1. Therefore, the graph must be a straight line. We can eliminate A and E, since neither are straight lines.
The equation is given in the slope-intercept form, 
, where 
stands for the slope of the line and 
stands for the line's y-intercept. Since 
, the coefficient of x, is positive here, we are looking for a line that goes upwards. We can eliminate D since it goes downwards, and therefore has a negative slope.
In the given equation, the constant 
, which represents the y-intercept, is also positive. Therefore, the line we are looking for also must intersect the y-axis at a positive value. The graph C appears to intersect the y-axis at a negative value, whereas the graph B appears to intersect the y-axis at a positive value. Therefore, B is the corresponding graph.
The power of 
The equation is given in the slope-intercept form, 
In the given equation, the constant 
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A restaurant sets the prices of its dishes using the following function:
Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5
where all quantities are in U.S. Dollars.
If the cost of ingredients for a steak dish is $14, what will the restaurant set the price of the steak dish at?
A restaurant sets the prices of its dishes using the following function:
Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5
where all quantities are in U.S. Dollars.
If the cost of ingredients for a steak dish is $14, what will the restaurant set the price of the steak dish at?
Tap to reveal answer
The price of the dish is a function of a single variable, the cost of the ingredients. One way to conceptualize the problem is by thinking of it in function notation. Let 
be the variable representing the cost of the ingredients. Let 
be a function of the cost of ingredients giving the price of the dish. Then, we can turn
"Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5"
into a regular equation with recognizable parts. Replace "Price" with 
and replace "cost of ingredients" with the variable 
.
Simplify by combining like terms (
and 
) to obtain:
The cost of ingredients for the steak dish is $14, so substitute 14 for 
.
All that's left is to compute the answer:
So, the steak dish will have a price of $24.60.
The price of the dish is a function of a single variable, the cost of the ingredients. One way to conceptualize the problem is by thinking of it in function notation. Let 
"Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5"
into a regular equation with recognizable parts. Replace "Price" with 
Simplify by combining like terms (
The cost of ingredients for the steak dish is $14, so substitute 14 for 
All that's left is to compute the answer:
So, the steak dish will have a price of $24.60.
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The daily pay in U.S. Dollars for a certain job is defined as the following function , where equals time in hours:
If an employee works for 7 hours in a day, how much is he or she paid?
The daily pay in U.S. Dollars for a certain job is defined as the following function
If an employee works for 7 hours in a day, how much is he or she paid?
Tap to reveal answer
The above function can be understood as, "An employee is paid $15 for each hour he or she works, plus a flat amount of $10."
If an employee works 7 hours, we can find the amount that he or she is paid by plugging in 7 for "Hours" in the equation:
Adhering to order of operations, we next find the product of 15 and 7:
Finally, we find the sum of 105 and 10:
The employee is paid $115.00 for seven hours' work.
The above function can be understood as, "An employee is paid $15 for each hour he or she works, plus a flat amount of $10."
If an employee works 7 hours, we can find the amount that he or she is paid by plugging in 7 for "Hours" in the equation:
Adhering to order of operations, we next find the product of 15 and 7:
Finally, we find the sum of 105 and 10:
The employee is paid $115.00 for seven hours' work.
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Consider the scenario below:
Helen is a painter. It takes her 3 days to make each painting. She has already made 6 paintings. Which of the following functions best models the number of paintings she will have after days?
Consider the scenario below:
Helen is a painter. It takes her 3 days to make each painting. She has already made 6 paintings. Which of the following functions best models the number of paintings she will have after
Tap to reveal answer
The question asks, "Which of the following functions best models the number of paintings she will have after days?"
From this, you know that the variable represents the number of days, and that represents the number of paintings she makes as a function of days spent working.
If it takes 3 days to make a painting, each day results in paintings. Therefore, we have a linear relationship with slope .
Additionally, she begins with 6 paintings. Therefore, even when zero days are spent working on paintings, she will have 6 paintings. In other words, . This means the y-intercept is 6.
As a result, the function will be
which can be rewritten as
The question asks, "Which of the following functions best models the number of paintings she will have after
From this, you know that the variable
If it takes 3 days to make a painting, each day results in
Additionally, she begins with 6 paintings. Therefore, even when zero days are spent working on paintings, she will have 6 paintings. In other words,
As a result, the function will be
which can be rewritten as
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The plot shows the graphs of five different equations. Using shape and location, determine which graphed line corresponds to the equation .
The plot shows the graphs of five different equations. Using shape and location, determine which graphed line corresponds to the equation
Tap to reveal answer
The power of and are both 1, so this is a linear equation of order 1. Therefore, the graph must be a straight line. We can eliminate A and E, since neither are straight lines.
The equation is given in the slope-intercept form, , where stands for the slope of the line and stands for the line's y-intercept. Since , the coefficient of x, is positive here, we are looking for a line that goes upwards. We can eliminate D since it goes downwards, and therefore has a negative slope.
In the given equation, the constant , which represents the y-intercept, is also positive. Therefore, the line we are looking for also must intersect the y-axis at a positive value. The graph C appears to intersect the y-axis at a negative value, whereas the graph B appears to intersect the y-axis at a positive value. Therefore, B is the corresponding graph.
The power of
The equation is given in the slope-intercept form,
In the given equation, the constant
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A restaurant sets the prices of its dishes using the following function:
Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5
where all quantities are in U.S. Dollars.
If the cost of ingredients for a steak dish is $14, what will the restaurant set the price of the steak dish at?
A restaurant sets the prices of its dishes using the following function:
Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5
where all quantities are in U.S. Dollars.
If the cost of ingredients for a steak dish is $14, what will the restaurant set the price of the steak dish at?
Tap to reveal answer
The price of the dish is a function of a single variable, the cost of the ingredients. One way to conceptualize the problem is by thinking of it in function notation. Let be the variable representing the cost of the ingredients. Let be a function of the cost of ingredients giving the price of the dish. Then, we can turn
"Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5"
into a regular equation with recognizable parts. Replace "Price" with and replace "cost of ingredients" with the variable .
Simplify by combining like terms ( and ) to obtain:
The cost of ingredients for the steak dish is $14, so substitute 14 for .
All that's left is to compute the answer:
So, the steak dish will have a price of $24.60.
The price of the dish is a function of a single variable, the cost of the ingredients. One way to conceptualize the problem is by thinking of it in function notation. Let
"Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5"
into a regular equation with recognizable parts. Replace "Price" with
Simplify by combining like terms (
The cost of ingredients for the steak dish is $14, so substitute 14 for
All that's left is to compute the answer:
So, the steak dish will have a price of $24.60.
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Define 
and 
.
Evaluate 
.
Define 
Evaluate 
Tap to reveal answer
By definition,
,
so first, evaluate 
by substituting 2 for 
in 
:
,
so evaluate 
through a similar substitution:
,
the correct response.
By definition,
so first, evaluate 
so evaluate 
the correct response.
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Define functions 
and 
as follows:
Evaluate 
.
Define functions 
Evaluate 
Tap to reveal answer
By definition, 
. Evaluate 
by substituting 0 for 
in the definition of 
:
, so substitute 4 for 
in the definition of 
. Since 4 satisfies the condition 
, use the definition for those values:
,
the correct value.
By definition, 
the correct value.
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Define functions 
and 
Give the domain of the composition 
.
Define functions 
Give the domain of the composition 
Tap to reveal answer
The domain of the composition of functions 
is that the set of all values 
that fall in the domain of 
such that 
falls in the domain of 
.
, a square root function, whose domain is the set of all values of 
that make the radicand nonnegative. Since the radicand is 
itself, the domain of 
is simply the set of all nonnegative numbers, or 
.
, a polynomial function, whose domain is the set of all real numbers. Therefore, the domain is not restricted further by 
. The domain of 
is 
.
The domain of the composition of functions 
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Define functions 
and 
.
Give the domain of the composition 
.
Define functions 
Give the domain of the composition 
Tap to reveal answer
The domain of the composition of functions 
is that the set of all values 
that fall in the domain of 
such that 
falls in the domain of 
.
, a rational function, so its domain is the set of all numbers except those that make its denominator zero. 
if and only if 
, making its domain 
.
, a square root function, so the domain of 
is the set of all values 
that make the radicand a nonnegative number. Since the radicand is 
itself, the domain of 
is simply the set of all nonnegative numbers, or 
.
Therefore, we need to further restrict the domain to those values 
in 
that make
Equivalently,
Since 1 is positive, and the quotient is nonnegative as well, it follows that
,
and
.
The domain of 
is therefore 
.
The domain of the composition of functions 
Therefore, we need to further restrict the domain to those values 
Equivalently,
Since 1 is positive, and the quotient is nonnegative as well, it follows that
and
The domain of 
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Define functions 
and 
.
Give the domain of the composition 
.
Define functions 
Give the domain of the composition 
Tap to reveal answer
The domain of the composition of functions 
is the set of all values 
that fall in the domain of 
such that 
falls in the domain of 
.
, a square root function, so the domain of 
is the set of all values 
that make the radicand a nonnegative number. Since the radicand is 
itself, the domain of 
is simply the set of all nonnegative numbers, or 
.
, a rational function, so its domain is the set of all numbers except those that make its denominator zero. 
if and only if 
, so from 
, we exclude only the value 
so that
;
that is,
Excluding this value from the domain, this leaves 
as the domain of 
.
The domain of the composition of functions 
that is,
Excluding this value from the domain, this leaves 
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Define functions 
and 
.
Give the domain of the composition 
.
Define functions 
Give the domain of the composition 
Tap to reveal answer
The domain of the composition of functions 
is that the set of all values 
that fall in the domain of 
such that 
falls in the domain of 
.
, a polynomial function, so its domain is the set of all real numbers.
, so the domain of 
is the set of all values 
that make the radicand a nonnegative number. Since the radicand is 
itself, the domain of 
is simply the set of all nonnegative numbers, or 
.
Therefore, the domain of 
is the set of all real numbers 
such that
.
Equivalently,
To solve this, first, solve the corresponding equation:
Factor the polynomial at left using the difference of squares pattern:
By the Zero Factor Property, either
, in which case 
,
or
, in which case 
.
These are the boundary values of the intervals to be tested. Choose an interior value of each interval and test it in the inequality:
Testing 
:
- True; include
Testing 
:
- False; exclude
Testing 
:
- True; include
Include 
and 
. Also include the boundary values, since equality is included in the inequality statement ( 
).
The correct domain is 
.
The domain of the composition of functions 
Therefore, the domain of 
Equivalently,
To solve this, first, solve the corresponding equation:
Factor the polynomial at left using the difference of squares pattern:
By the Zero Factor Property, either
or
These are the boundary values of the intervals to be tested. Choose an interior value of each interval and test it in the inequality:
Testing 
Testing 
Testing 
Include 
The correct domain is 
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Define functions 
and 
.
Give the domain of the composition 
.
Define functions
Give the domain of the composition
Tap to reveal answer
The domain of the composition of functions 
is the set of all values 
that fall in the domain of 
such that 
falls in the domain of 
.
, a square root function, so the domain of 
is the set of all values 
that make the radicand a nonnegative number. Since the radicand is 
itself, the domain of 
is simply the set of all nonnegative numbers, or 
.
, a polynomial function; its domain is the set of all reals, so it does not restrict the domain any further.
The correct domain is .
The domain of the composition of functions
The correct domain is
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Define functions 
and 
.
Give the domain of the function 
.
Define functions 
Give the domain of the function 
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The domain of the sum of two functions is the intersections of the domains of the individual functions. Both 
and 
are square root functions, so their radicands must both be positive.
The domain of 
is the set of all values of 
such that:
Adding 8 to both sides:
This domain is 
.
The domain of 
is the set of all values of 
such that:
Subtracting 8 tfrom both sides:
This domain is 
The domain of 
is the intersection of the domains, which is 
.
The domain of the sum of two functions is the intersections of the domains of the individual functions. Both 
The domain of 
Adding 8 to both sides:
This domain is 
The domain of 
Subtracting 8 tfrom both sides:
This domain is
The domain of 
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Define functions 
and 
.
Give the domain of the function 
.
Define functions 
Give the domain of the function 
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The domain of the difference of two functions is the intersections of the domains of the individual functions. Both 
and 
are square root functions, so their radicands must both be positive.
The domain of 
is the set of all values of 
such that:
Subtracting 8 tfrom both sides:
This domain is 
The domain of 
is the set of all values of 
such that:
Adding 
to both sides:
, or
This domain is 
.
The domain of 
is the intersection of these, 
.
The domain of the difference of two functions is the intersections of the domains of the individual functions. Both 
The domain of 
Subtracting 8 tfrom both sides:
This domain is
The domain of 
Adding 
This domain is 
The domain of 
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Define functions 
and 
.
Give the domain of the function 
.
Define functions
Give the domain of the function
Tap to reveal answer
The domain of the sum of two functions is the intersections of the domains of the individual functions.
Both 
and 
are cube root functions. There is no restriction on the value of the radicand of a cube root, so both of these functions have as their domain the set of all real numbers 
. It follows that 
also has domain 
.
The domain of the sum of two functions is the intersections of the domains of the individual functions.
Both
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