Algebraic Functions - ACT Math
Card 0 of 20
Define function 
as follows:
On which of the following restrictions of the domain of 
would 
not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain 
such that 
.
is a quadratic function, so its graph is a parabola. The key is to find the 
-intercept of the vertex of the parabola, which can be found by completing the square:
The vertex happens at 
, so the interval which contains this value will have at least one pair 
such that 
. The correct choice is 
.
The vertex happens at
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Define function 
as follows:
On which of the following restrictions of the domain of would not exist?
Define function
On which of the following restrictions of the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain 
such that 
.
has a sinusoidal wave as its graph, with period 
; it begins at a relative maximum of 
and has a relative maximum or minimum every 
units. Therefore, any interval containing an integer multiple of 
will have at least two distinct values 
such that 
.
The only interval among the choices that includes a multiple of 
is 
:
.
This is the correct choice.
The only interval among the choices that includes a multiple of
This is the correct choice.
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Define function 
as follows:
In which of the following ways could the domain of 
be restricted so that 
does not have an inverse?
Define function
In which of the following ways could the domain of
If 
, then 
. By the addition property of inequality, if 
, then 
. Therefore, if 
, 
.
Consequently, there can be no 
such that 
, regardless of how the domain is restricted. 
will have an inverse regardless of any domain restriction.
If
Consequently, there can be no
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Define function 
as follows:
Suppose the domain of 
were to be restricted so that 
could have an inverse. Which of the following restrictions would not give 
an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain 
such that 
.
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are 
.
has one boundary that is a zero and one interior point that is a zero. Therefore, there is a vertex in the interior of the interval, so it will have at least one pair such that . Since a cubic polynomial has two "arms", one going up and one going down, 
will increase as 
increases in the other four intervals. 
is the correct choice.
The key to this question is to find the zeroes of the polynomial, which can be done as follows:
'
The zeroes are
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Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
Consider the following statement to be true:
If a fish is a carnivore, then it is a shark.
Which of the following statements must also be true?
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").
Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").
Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").
Answer choice D states "If not Y, then X", which is false.
Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.
Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."
The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.
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Define function 
as follows:
Suppose the domain of 
were to be restricted so that 
could have an inverse. Which of the following restrictions would not give 
an inverse?
Define function
Suppose the domain of
has an inverse on a given domain if and only if there are no two distinct values on the domain such that .
has a sinusoidal wave as its graph, with period and phase shift 
units to the left. Its positive "peaks" and "valleys" begin at 
and occur every 
units.
Since 
includes one of these "peaks" or "valleys", it contains at least two distinct values such that . It is the correct choice.
Since
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A function f(x) = –1 for all values of x. Another function g(x) = 3_x_ for all values of x. What is g(f(x)) when x = 4?
A function f(x) = –1 for all values of x. Another function g(x) = 3_x_ for all values of x. What is g(f(x)) when x = 4?
We work from the inside out, so we start with the function f(x). f(4) = –1. Then we plug that value into g(x), so g(f(x)) = 3 * (–1) = –3.
We work from the inside out, so we start with the function f(x). f(4) = –1. Then we plug that value into g(x), so g(f(x)) = 3 * (–1) = –3.
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What is f(–3) if f(x) = _x_2 + 5?
What is f(–3) if f(x) = _x_2 + 5?
f(–3) = (–3)2 + 5 = 9 + 5 = 14
f(–3) = (–3)2 + 5 = 9 + 5 = 14
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For all values of x, f(x) = 7_x_2 – 3, and for all values of y, g(y) = 2_y_ + 9. What is g(f(x))?
For all values of x, f(x) = 7_x_2 – 3, and for all values of y, g(y) = 2_y_ + 9. What is g(f(x))?
The inner function f(x) is like our y-value that we plug into g(y).
g(f(x)) = 2(7_x_2 – 3) + 9 = 14_x_2 – 6 + 9 = 14_x_2 + 3.
The inner function f(x) is like our y-value that we plug into g(y).
g(f(x)) = 2(7_x_2 – 3) + 9 = 14_x_2 – 6 + 9 = 14_x_2 + 3.
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g(x) = 4x – 3
h(x) = .25πx + 5
If f(x)=g(h(x)). What is f(1)?
g(x) = 4x – 3
h(x) = .25πx + 5
If f(x)=g(h(x)). What is f(1)?
First, input the function of h into g. So f(x) = 4(.25πx + 5) – 3, then simplify this expression f(x) = πx + 20 – 3 (leave in terms of πsince our answers are in terms of π). Then plug in 1 for x to get π+ 17.
First, input the function of h into g. So f(x) = 4(.25πx + 5) – 3, then simplify this expression f(x) = πx + 20 – 3 (leave in terms of πsince our answers are in terms of π). Then plug in 1 for x to get π+ 17.
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If F(x) = 2x2 + 3 and G(x) = x – 3, what is F(G(x))?
If F(x) = 2x2 + 3 and G(x) = x – 3, what is F(G(x))?
A composite function substitutes one function into another function and then simplifies the resulting expression. F(G(x)) means the G(x) gets put into F(x).
F(G(x)) = 2(x – 3)2 + 3 = 2(x2 – 6x +9) + 3 = 2x2 – 12x + 18 + 3 = 2x2 – 12x + 21
G(F(x)) = (2x2 +3) – 3 = 2x2
A composite function substitutes one function into another function and then simplifies the resulting expression. F(G(x)) means the G(x) gets put into F(x).
F(G(x)) = 2(x – 3)2 + 3 = 2(x2 – 6x +9) + 3 = 2x2 – 12x + 18 + 3 = 2x2 – 12x + 21
G(F(x)) = (2x2 +3) – 3 = 2x2
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If a(x) = 2x3 + x, and b(x) = –2x, what is a(b(2))?
If a(x) = 2x3 + x, and b(x) = –2x, what is a(b(2))?
When functions are set up within other functions like in this problem, the function closest to the given variable is performed first. The value obtained from this function is then plugged in as the variable in the outside function. Since b(x) = –2x, and x = 2, the value we obtain from b(x) is –4. We then plug this value in for x in the a(x) function. So a(x) then becomes 2(–43) + (–4), which equals –132.
When functions are set up within other functions like in this problem, the function closest to the given variable is performed first. The value obtained from this function is then plugged in as the variable in the outside function. Since b(x) = –2x, and x = 2, the value we obtain from b(x) is –4. We then plug this value in for x in the a(x) function. So a(x) then becomes 2(–43) + (–4), which equals –132.
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Let F(x) = _x_3 + 2_x_2 – 3 and G(x) = x + 5. Find F(G(x))
Let F(x) = _x_3 + 2_x_2 – 3 and G(x) = x + 5. Find F(G(x))
F(G(x)) is a composite function where the expression G(x) is substituted in for x in F(x)
F(G(x)) = (x + 5)3 + 2(x + 5)2 – 3 = x_3 + 17_x_2 + 95_x + 172
G(F(x)) = _x_3 + _x_2 + 2
F(x) – G(x) = _x_3 + 2_x_2 – x – 8
F(x) + G(x) = _x_3 + 2_x_2 + x + 2
F(G(x)) is a composite function where the expression G(x) is substituted in for x in F(x)
F(G(x)) = (x + 5)3 + 2(x + 5)2 – 3 = x_3 + 17_x_2 + 95_x + 172
G(F(x)) = _x_3 + _x_2 + 2
F(x) – G(x) = _x_3 + 2_x_2 – x – 8
F(x) + G(x) = _x_3 + 2_x_2 + x + 2
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What is the value of xy_2(xy – 3_xy) given that x = –3 and y = 7?
What is the value of xy_2(xy – 3_xy) given that x = –3 and y = 7?
Evaluating yields –6174.
–147(–21 + 63) =
–147 * 42 = –6174
Evaluating yields –6174.
–147(–21 + 63) =
–147 * 42 = –6174
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Given the functions f(x) = 2_x_ + 4 and g(x) = 3_x_ – 6, what is f(g(x)) when x = 6?
Given the functions f(x) = 2_x_ + 4 and g(x) = 3_x_ – 6, what is f(g(x)) when x = 6?
We need to work from the inside to the outside, so g(6) = 3(6) – 6 = 12.
Then f(g(6)) = 2(12) + 4 = 28.
We need to work from the inside to the outside, so g(6) = 3(6) – 6 = 12.
Then f(g(6)) = 2(12) + 4 = 28.
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If f(x)=3x and g(x)=2x+2, what is the value of f(g(x)) when x=3?
If f(x)=3x and g(x)=2x+2, what is the value of f(g(x)) when x=3?
With composition of functions (as with the order of operations) we perform what is inside of the parentheses first. So, g(3)=2(3)+2=8 and then f(8)=24.
With composition of functions (as with the order of operations) we perform what is inside of the parentheses first. So, g(3)=2(3)+2=8 and then f(8)=24.
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The function 
is defined as 
. What is 
?
The function 
Substitute -1 for 
in the given function.
If you didn’t remember the negative sign, you will have calculated 36. If you remembered the negative sign at the very last step, you will have calculated -36; however, if you did not remember that 
is 1, then you will have calculated 18.
Substitute -1 for 
If you didn’t remember the negative sign, you will have calculated 36. If you remembered the negative sign at the very last step, you will have calculated -36; however, if you did not remember that 
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What is the value of the function f(x) = 6x2 + 16x – 6 when x = –3?
What is the value of the function f(x) = 6x2 + 16x – 6 when x = –3?
There are two ways to do this problem. The first way just involves plugging in –3 for x and solving 6〖(–3)〗2 + 16(–3) – 6, which equals 54 – 48 – 6 = 0. The second way involves factoring the polynomial to (6x – 2)(x + 3) and then plugging in –3 for x. The second way quickly shows that the answer is 0 due to multiplying by (–3 + 3).
There are two ways to do this problem. The first way just involves plugging in –3 for x and solving 6〖(–3)〗2 + 16(–3) – 6, which equals 54 – 48 – 6 = 0. The second way involves factoring the polynomial to (6x – 2)(x + 3) and then plugging in –3 for x. The second way quickly shows that the answer is 0 due to multiplying by (–3 + 3).
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If 
and 
, what is 
?
If
Plug g(x) into f(x) as if it is just a variable. This gives f(g(x)) = 3(x2 – 12) + 7.
Distribute the 3: 3x2 – 36 + 7 = 3x2 – 29
Plug g(x) into f(x) as if it is just a variable. This gives f(g(x)) = 3(x2 – 12) + 7.
Distribute the 3: 3x2 – 36 + 7 = 3x2 – 29
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If the function 
is created by shifting 
up four units and then reflecting it across the x-axis, which of the following represents 
in terms of 
?
If the function 
We can take each of the listed transformations of 
one at a time. If 
is to be shifted up by four units, increase every value of 
by 4.
Next, take this equation and reflect it across the x-axis. If we reflect a function across the x-axis, then all of its values will be multiplied by negative one. So, 
can be written in the following way:
Lastly, distribute the negative sign to arrive at the final answer.
We can take each of the listed transformations of 
Next, take this equation and reflect it across the x-axis. If we reflect a function across the x-axis, then all of its values will be multiplied by negative one. So, 
Lastly, distribute the negative sign to arrive at the final answer.
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