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ACT Math : Algebraic Functions

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Example Questions

Example Question #41 : Algebraic Functions

An outpost has the supplies to last 2 people for 14 days. How many days will the supplies last for 7 people?

Possible Answers:

$\dpi{100} \small 9$

$\dpi{100} \small 4$

$\dpi{100} \small 7$

$\dpi{100} \small 5$

$\dpi{100} \small 10$

Correct answer:

$\dpi{100} \small 4$

Explanation:

Supplies are used at the rate of $\dpi{100} \small \frac{Supplies}{Days\times People}$ .

Since the total amount of supplies is the same in either case, $\dpi{100} \small \frac{1}{14\times 2}=\frac{1}{7\times \ (\&hash;\ of\ days)}$ .

Solve for days to find that the supplies will last for 4 days.

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Example Question #23 : Algebraic Functions

Worker $\dpi{100} \small A$ can make a trinket in 4 hours, Worker $\dpi{100} \small B$ can make a trinket in 2 hours. When they work together, how long will it take them to make a trinket?

Possible Answers:

$\dpi{100} \small \frac{1}{2}\ hour$

$\dpi{100} \small \ 1 \frac{1}{3}\ hours$

$\dpi{100} \small 6\ hours$

$\dpi{100} \small \ 1 \frac{1}{2}\ hours$

$\dpi{100} \small 3\ hours$

Correct answer:

$\dpi{100} \small \ 1 \frac{1}{3}\ hours$

Explanation:

The rates are what needs to be added. Rate $\dpi{100} \small A$ is $\dpi{100} \small \frac{1}{4}$ , or one trinket every 4 hours. Rate $\dpi{100} \small B$ is $\dpi{100} \small \frac{1}{2}$ , one per two hours.

$\dpi{100} \small \frac{1}{4}+ \frac{1}{2}=\frac{3}{4}$ , their combined rate in trinkets per hour.

Now invert the equation to get back to hours per trinket, which is what the question asks for: $\dpi{100} \small \frac{4}{3}\ or \ 1 \frac{1}{3}$

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Example Question #43 : Algebraic Functions

$a\ \mho \ b = a(b+1)-3$

Quantity A Quantity B

$1\ \mho\ 1$ $2\ \mho\ 0$

Possible Answers:

Quantity A and Quantity B are equal

Quantity A is greater

Quantity B is greater

The relationship cannot be determined from the information given.

Correct answer:

Quantity A and Quantity B are equal

Explanation:

Since $a\ \mho \ b = a(b+1)-3$ , then we have that

$1\ \mho \ 1=1(1+1)-3=-1$

and

$2\ \mho \ 0=2(0+1)-3=-1$ .

Thus, the two quantities are equal.

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Example Question #44 : Algebraic Functions

If the average of two numbers is $\dpi{100} \small 3y$ and one of the numbers is $\dpi{100} \small y+z$ , what is the other number, in terms of $\dpi{100} \small y$ and $\dpi{100} \small z$ ?

Possible Answers:

$\dpi{100} \small 3y+z$

$\dpi{100} \small 5y-z$

$\dpi{100} \small 5y+z$

$\dpi{100} \small y+z$

$\dpi{100} \small 4y-z$

Correct answer:

$\dpi{100} \small 5y-z$

Explanation:

The average is the sum of the terms divided by the number of terms. Here you have $\dpi{100} \small y+z$ and the other number which you can call $\dpi{100} \small x$ . The average of $\dpi{100} \small x$ and $\dpi{100} \small y+z$ is $\dpi{100} \small 3y$ . So $\dpi{100} \small 3y=\frac{(x+y+z)}{2}$

Multiply both sides by 2.

Solve for $\dpi{100} \small x=5y-z$ .

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Example Question #45 : Algebraic Functions

Alice is twice as old as Tom, but four years ago, she was three years older than Tom is now. How old is Tom now?

Possible Answers:

$\dpi{100} \small 13$

$\dpi{100} \small 21$

$\dpi{100} \small 3$

$\dpi{100} \small 7$

$\dpi{100} \small 9$

Correct answer:

$\dpi{100} \small 7$

Explanation:

The qustion can be broken into two equations with two unknows, Alice age $\dpi{100} \small (A)$ and Tom's age $\dpi{100} \small (T)$ .

$\dpi{100} \small A=2T$

$\dpi{100} \small A-4=T+3$

$\dpi{100} \small 2T-4=T+3$

$\dpi{100} \small T=7$

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Example Question #46 : Algebraic Functions

A jet goes from City 1 to City 2 at an average speed of 600 miles per hour, and returns along the same path at an average speed if 300 miles per hour. What is the average speed, in miles per hour, for the trip?

Possible Answers:

450miles/hour

400miles/hour

350miles/hour

300miles/hour

500miles/hour

Correct answer:

400miles/hour

Explanation:

Chose a number for the distance between City 1 and 2; 1800 works well, as it is a multiple of 600 and 300.

Now, find the time for each trip, the total distance, and the total time.

$t_1=\frac{d_1}{v_1}=\frac{1800}{600}=3$

$t_2=\frac{d_2}{v_2}=\frac{1800}{300}=6$

$total\ t=t_1+t_2=3+6=9$

$total\ d=d_1+d_2=1800+1800=3600$

Now we can find the average speed by dividing the total distance by the total time.

$average speed = \frac{total distance}{total time} = \frac{3600 miles}{9 hours} = 400 miles/hour$

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Example Question #47 : Algebraic Functions

$f(x)=3x^{2}-5$

g(x)=9-2x

Find f(g(5)) .

Possible Answers:

131

Correct answer:

Explanation:

Plug 5 into g(x) first:

g(5)=9-2(5)

=-1

Now, plug this answer into f(x) :

$f(-1)=3(-1)^{2}-5$

=3-5

=-2

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Example Question #48 : Algebraic Functions

If f(x) = 3x + 7 and g(x) = x^2 - 12 , what is f(g(x)) ?

Possible Answers:

3x^2 + 29

3x^3 - 29

3x^2 - 29

9x^2 - 29

9x^3 + 29

Correct answer:

3x^2 - 29

Explanation:

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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Example Question #31 : How To Find F(X)

If h(x)=2x+3 and g(x) = x-4 , then g(h(3))=?

Possible Answers:

Correct answer:

Explanation:

To answer this question, we need to understand exactly what g(h(3)) actually means.

We start from the inside of the parentheses and work outwards. Therefore, we first solve for h(3) using the equation for h(x) provided for us. So, for this data:

h(3)=h(x)=2x+3= 2(3)+3=6+3=9

Therefore, h(3)=9 . We now take that answer and plug it in for the value within g(x) = x-4 . So, for this data:

g(x) = x-4=9-4=5

Therefore, the answer to g(h(3)) is .

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Example Question #47 : Algebraic Functions

If f(x) = 2x^2 + 3x + 2 , what does f(x + 4) equal?

Possible Answers:

2x^2 + 11x + 30

2x^2 + 19x + 46

2x^2 + 12x + 56

2x^2 + 10x + 42

2x^2 + 19x + 38

Correct answer:

2x^2 + 19x + 46

Explanation:

For a question like this, treat it just like you would the use of a numeric value for evaluating your function. All you do is “plug in” x + 4 . Thus, for this function, you get:

f(x + 4) = 2(x + 4)^2 + 3(x+4) + 2

Next, you just need to distribute everything correctly:

f(x + 4) = 2(x + 4)(x + 4) + 3x+12 + 2

f(x + 4) = 2(x^2 + 8x + 16) + 3x + 14

f(x + 4) = 2x^2 + 16x + 32 + 3x + 14

f(x + 4) = 2x^2 + 19x + 46

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ACT Math : Algebraic Functions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #41 : Algebraic Functions

Example Question #23 : Algebraic Functions

Example Question #43 : Algebraic Functions

Example Question #44 : Algebraic Functions

Example Question #45 : Algebraic Functions

Example Question #46 : Algebraic Functions

Example Question #47 : Algebraic Functions

Example Question #48 : Algebraic Functions

Example Question #31 : How To Find F(X)

Example Question #47 : Algebraic Functions

All ACT Math Resources

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