How to find f(x) - PSAT Math
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The rate of a gym membership costs p dollars the first month and m dollars per month every month thereafter. Which of the following represents the total cost of the gym membership for n months, if n is a positive integer?
The rate of a gym membership costs p dollars the first month and m dollars per month every month thereafter. Which of the following represents the total cost of the gym membership for n months, if n is a positive integer?
The one-time first-month cost is p, and the monthly cost is m, which gets multipled by every month but the first (of which there are n -1). The total cost is the first-month cost of p, plus the monthly cost for (i.e. times) n -1 months, which makes the total cost equal to p + m (n -1).
The one-time first-month cost is p, and the monthly cost is m, which gets multipled by every month but the first (of which there are n -1). The total cost is the first-month cost of p, plus the monthly cost for (i.e. times) n -1 months, which makes the total cost equal to p + m (n -1).
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- If f(x) = (x + 4)/(x – 4) for all integers except x = 4, which of the following has the lowest value?
- If f(x) = (x + 4)/(x – 4) for all integers except x = 4, which of the following has the lowest value?
Plug each value for x into the above equation and solve for f(x). f(1) provides the lowest value –5/3
Plug each value for x into the above equation and solve for f(x). f(1) provides the lowest value –5/3
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If n and p are positive and 100_n_3_p_-1 = 25_n_, what is n-2 in terms of p ?
If n and p are positive and 100_n_3_p_-1 = 25_n_, what is n-2 in terms of p ?
To solve this problem, we look for an operation to perform on both sides that will leave n-2 by itself on one side. Dividing both sides by 25_n_-3 would leave n-2 by itself on the right side of the equqation, as shown below:
100n3p–1/25n–3 = 25n/25n–3
Remember that when dividing terms with the same base, we subtract the exponents, so the equation can be written as 100n0p–1/25 = n–2
Finally, we simplify to find 4_p–_1 = _n–_2.
To solve this problem, we look for an operation to perform on both sides that will leave n-2 by itself on one side. Dividing both sides by 25_n_-3 would leave n-2 by itself on the right side of the equqation, as shown below:
100n3p–1/25n–3 = 25n/25n–3
Remember that when dividing terms with the same base, we subtract the exponents, so the equation can be written as 100n0p–1/25 = n–2
Finally, we simplify to find 4_p–_1 = _n–_2.
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If 7y = 4x - 12, then x =
If 7y = 4x - 12, then x =
Adding 12 to both sides and dividing by 4 yields (7y+12)/4.
Adding 12 to both sides and dividing by 4 yields (7y+12)/4.
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Which of the statements describes the solution set for **–**7(x + 3) = **–**7x + 20 ?
Which of the statements describes the solution set for **–**7(x + 3) = **–**7x + 20 ?
By distribution we obtain **–**7x – 21 = – 7x + 20. This equation is never possibly true.
By distribution we obtain **–**7x – 21 = – 7x + 20. This equation is never possibly true.
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A function F is defined as follows:
for x2 > 1, F(x) = 4x2 + 2x – 2
for x2 < 1, F(x) = 4x2 – 2x + 2
What is the value of F(1/2)?
A function F is defined as follows:
for x2 > 1, F(x) = 4x2 + 2x – 2
for x2 < 1, F(x) = 4x2 – 2x + 2
What is the value of F(1/2)?
For F(1/2), x2=1/4, which is less than 1, so we use the bottom equation to solve. This gives F(1/2)= 4(1/2)2 – 2(1/2) + 2 = 1 – 1 + 2 = 2
For F(1/2), x2=1/4, which is less than 1, so we use the bottom equation to solve. This gives F(1/2)= 4(1/2)2 – 2(1/2) + 2 = 1 – 1 + 2 = 2
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For which value of 
are the following two functions equal?
For which value of 
It is important to follow the order of operations for this equation and find a solution that satisfies both F(x) and G(x).
Recall the order of operations is PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
The correct answer is 4 because
F(x) = 2_x_ + 3_x_ + (9_x_/3) = 2(4) + 34 + ((9 * 4)/3) = 101, and
G(x) = (((24 + 44)/2) - 4 * 4) – 5(4) + 1 = 101.
It is important to follow the order of operations for this equation and find a solution that satisfies both F(x) and G(x).
Recall the order of operations is PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
The correct answer is 4 because
F(x) = 2_x_ + 3_x_ + (9_x_/3) = 2(4) + 34 + ((9 * 4)/3) = 101, and
G(x) = (((24 + 44)/2) - 4 * 4) – 5(4) + 1 = 101.
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Given f(x)=|3x-2|. What values of x satisfy f(x)=10
Given f(x)=|3x-2|. What values of x satisfy f(x)=10
Setting f(x)=10 and taking the equation out of the absolute value you get 10=3x-2 and -10=3x-2. Solving both of these equations for x gives you x=4 or -8/3.
Setting f(x)=10 and taking the equation out of the absolute value you get 10=3x-2 and -10=3x-2. Solving both of these equations for x gives you x=4 or -8/3.
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If f(x) = x² – 2 and g(x) = 3x + 5, what is f(g(x))?
If f(x) = x² – 2 and g(x) = 3x + 5, what is f(g(x))?
To find f(g(x) plug the equation for g(x) into equation f(x) in place of “x” so that you have: f(g(x)) = (3x + 5)² – 2.
Simplify: (3x + 5)(3x + 5) – 2
Use FOIL: 9x² + 30x + 25 – 2 = 9x² + 30x + 23
To find f(g(x) plug the equation for g(x) into equation f(x) in place of “x” so that you have: f(g(x)) = (3x + 5)² – 2.
Simplify: (3x + 5)(3x + 5) – 2
Use FOIL: 9x² + 30x + 25 – 2 = 9x² + 30x + 23
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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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Let f(x, y) = x2y2 – xy + y. If a = f(1, 3), and b = f(–2, –1), then what is f(a, b)?
Let f(x, y) = x2y2 – xy + y. If a = f(1, 3), and b = f(–2, –1), then what is f(a, b)?
f(x, y) is defined as x2y2 – xy + y. In order to find f(a, b), we will need to first find a and then b.
We are told that a = f(1, 3). We can use the definition of f(x, y) to determine the value of a.
a = f(1, 3) = 1232 – 1(3) + 3 = 1(9) – 3 + 3 = 9 + 0 = 9
a = 9
Similarly, we can find b by determining the value of f(–2, –1).
b = f(–2, –1) = (–2)2(–1)2 – (–2)(–1) + –1 = 4(1) – (2) – 1 = 4 – 2 – 1 = 1
b = 1
Now, we can find f(a, b), which is equal to f(9, 1).
f(a, b) = f(9, 1) = 92(12) – 9(1) + 1 = 81 – 9 + 1 = 73
f(a, b) = 73
The answer is 73.
f(x, y) is defined as x2y2 – xy + y. In order to find f(a, b), we will need to first find a and then b.
We are told that a = f(1, 3). We can use the definition of f(x, y) to determine the value of a.
a = f(1, 3) = 1232 – 1(3) + 3 = 1(9) – 3 + 3 = 9 + 0 = 9
a = 9
Similarly, we can find b by determining the value of f(–2, –1).
b = f(–2, –1) = (–2)2(–1)2 – (–2)(–1) + –1 = 4(1) – (2) – 1 = 4 – 2 – 1 = 1
b = 1
Now, we can find f(a, b), which is equal to f(9, 1).
f(a, b) = f(9, 1) = 92(12) – 9(1) + 1 = 81 – 9 + 1 = 73
f(a, b) = 73
The answer is 73.
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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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The cost of a cell phone plan is \$40 for the first 100 minutes of calls, and then 5 cents for each minute after. If the variable x is equal to the number of minutes used for calls in a month on that cell phone plan, what is the equation f(x) for the cost, in dollars, of the cell phone plan for calls during that month?
The cost of a cell phone plan is \$40 for the first 100 minutes of calls, and then 5 cents for each minute after. If the variable x is equal to the number of minutes used for calls in a month on that cell phone plan, what is the equation f(x) for the cost, in dollars, of the cell phone plan for calls during that month?
40 dollars is the constant cost of the cell phone plan, regardless of minute usage for calls. We then add 5 cents, or 0.05 dollars, for every minute of calls over 100. Thus, we do not multiply 0.05 by x, but rather by (x - 100), since the 5 cent charge only applies to minutes used that are over the 100-minute barrier. For example, if you used 101 minutes for calls during the month, you would only pay the 5 cents for that 101st minute, making your cost for calls \$40.05. Thus, the answer is 40 + 0.05(x - 100).
40 dollars is the constant cost of the cell phone plan, regardless of minute usage for calls. We then add 5 cents, or 0.05 dollars, for every minute of calls over 100. Thus, we do not multiply 0.05 by x, but rather by (x - 100), since the 5 cent charge only applies to minutes used that are over the 100-minute barrier. For example, if you used 101 minutes for calls during the month, you would only pay the 5 cents for that 101st minute, making your cost for calls \$40.05. Thus, the answer is 40 + 0.05(x - 100).
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f(x) = 4x + 2
g(x) = 3x - 1
The two equations above define the functions f(x) = g(x). If f(d) = 2g(d) for some value of d, then what is the value of d?
f(x) = 4x + 2
g(x) = 3x - 1
The two equations above define the functions f(x) = g(x). If f(d) = 2g(d) for some value of d, then what is the value of d?
f(x) = 4x + 2
g(x) = 3x - 1
We have f(d) = 2g(d). We multiply each value in g(d) by 2.
4d + 2 = 2(3d - 1) (Distribute the 2 in the parentheses by multiplying each value in them by 2.)
4d + 2 = 6d - 2 (Add 2 to both sides.)
4d + 4 = 6d (Subtract 4d from both sides.)
4 = 2d (Divide both sides by 2.)
2 = d
We can plug that back in to double check.
4(2) + 2 = 6(2) - 2
8 + 2 = 12 - 2
10 = 10
f(x) = 4x + 2
g(x) = 3x - 1
We have f(d) = 2g(d). We multiply each value in g(d) by 2.
4d + 2 = 2(3d - 1) (Distribute the 2 in the parentheses by multiplying each value in them by 2.)
4d + 2 = 6d - 2 (Add 2 to both sides.)
4d + 4 = 6d (Subtract 4d from both sides.)
4 = 2d (Divide both sides by 2.)
2 = d
We can plug that back in to double check.
4(2) + 2 = 6(2) - 2
8 + 2 = 12 - 2
10 = 10
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The function f, where f(x) = x2 + 6x + 8, is related to function g, where g(x) = 5 f(x-2). What is g(3)?
The function f, where f(x) = x2 + 6x + 8, is related to function g, where g(x) = 5 f(x-2). What is g(3)?
Doing things in an orderly way is a friend to the test-taker.
g(3) = 5 f(3-2)
= 5 f(1)
= 5 \[ 12 + 6**∙**1 + 8\]
= 5 \[ 1 + 6 + 8\]
= 5 \[ 15\]
= 75
Doing things in an orderly way is a friend to the test-taker.
g(3) = 5 f(3-2)
= 5 f(1)
= 5 \[ 12 + 6**∙**1 + 8\]
= 5 \[ 1 + 6 + 8\]
= 5 \[ 15\]
= 75
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If f(x) = 5x – 10, then what is the value of 5(f(10)) – 10?
If f(x) = 5x – 10, then what is the value of 5(f(10)) – 10?
The first step is to find what f(10) equals, so f(10)=5(10) – 10 = 40. Then substitute 40 into the second equation: 5(40) – 10 = 200 – 10 = 190.
190 is the correct answer
The first step is to find what f(10) equals, so f(10)=5(10) – 10 = 40. Then substitute 40 into the second equation: 5(40) – 10 = 200 – 10 = 190.
190 is the correct answer
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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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If f(x) = _x_2 – 5 for all values x and f(a) = 4, what is one possible value of a?
If f(x) = _x_2 – 5 for all values x and f(a) = 4, what is one possible value of a?
Using the defined function, f(a) will produce the same result when substituted for x:
f(a) = _a_2 – 5
Setting this equal to 4, you can solve for a:
_a_2 – 5 = 4
_a_2 = 9
a = –3 or 3
Using the defined function, f(a) will produce the same result when substituted for x:
f(a) = _a_2 – 5
Setting this equal to 4, you can solve for a:
_a_2 – 5 = 4
_a_2 = 9
a = –3 or 3
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If the function g is defined by g(x) = 4_x_ + 5, then 2_g_(x) – 3 =
If the function g is defined by g(x) = 4_x_ + 5, then 2_g_(x) – 3 =
The function g(x) is equal to 4_x_ + 5, and the notation 2_g_(x) asks us to multiply the entire function by 2. 2(4_x_ + 5) = 8_x_ + 10. We then subtract 3, the second part of the new equation, to get 8_x_ + 7.
The function g(x) is equal to 4_x_ + 5, and the notation 2_g_(x) asks us to multiply the entire function by 2. 2(4_x_ + 5) = 8_x_ + 10. We then subtract 3, the second part of the new equation, to get 8_x_ + 7.
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If f(x) = x_2 + 5_x and g(x) = 2, what is f(g(4))?
If f(x) = x_2 + 5_x and g(x) = 2, what is f(g(4))?
First you must find what g(4) is. The definition of g(x) tells you that the function is always equal to 2, regardless of what “x” is. Plugging 2 into f(x), we get 22 + 5(2) = 14.
First you must find what g(4) is. The definition of g(x) tells you that the function is always equal to 2, regardless of what “x” is. Plugging 2 into f(x), we get 22 + 5(2) = 14.
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