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ACT Math : Algebra

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Example Questions

Example Question #2791 : Sat Mathematics

Let F(x) = x³ + 2x² – 3 and G(x) = x + 5. Find F(G(x))

Possible Answers:

x³ + x² + x + 8

x³ + x² + 2

x³ + 2x² + x + 2

x³ + 17x² + 95x + 172

x³ + 2x² – x – 8

Correct answer:

x³ + 17x² + 95x + 172

Explanation:

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 = x³ + 17x² + 95x + 172

G(F(x)) = x³ + x² + 2

F(x) – G(x) = x³ + 2x² – x – 8

F(x) + G(x) = x³ + 2x² + x + 2

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Example Question #2792 : Sat Mathematics

What is the value of xy²(xy – 3xy) given that x = –3 and y = 7?

Possible Answers:

–2881

3565

–6174

2881

Correct answer:

–6174

Explanation:

Evaluating yields –6174.

–147(–21 + 63) =

–147 * 42 = –6174

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Example Question #2792 : Sat Mathematics

$f(x)=x^{2}+2$ $\displaystyle f(x)=x^{2}+2$

$g(x)=x-4$ $\displaystyle g(x)=x-4$

Find $g(f(2))$ $\displaystyle g(f(2))$ .

Possible Answers:

$\dpi{100} \small 4$ $\displaystyle \dpi{100} \small 4$

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

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

$\dpi{100} \small 2$ $\displaystyle \dpi{100} \small 2$

$\dpi{100} \small 1$ $\displaystyle \dpi{100} \small 1$

Correct answer:

$\dpi{100} \small 2$ $\displaystyle \dpi{100} \small 2$

Explanation:

$g(f(2))$ $\displaystyle g(f(2))$ is $\dpi{100} \small 2$ $\displaystyle \dpi{100} \small 2$ . To start, we find that $f(2)=2^{2}+2=4+2=6$ $\displaystyle f(2)=2^{2}+2=4+2=6$ . Using this, we find that $g(6)=6-4=2$ $\displaystyle g(6)=6-4=2$ .

Alternatively, we can find that $g(f(x))=(x^{2}+2)-4=x^{2}-2$ $\displaystyle g(f(x))=(x^{2}+2)-4=x^{2}-2$ . Then, we find that $g(f(2))=2^{2}-2=4-2=2$ $\displaystyle g(f(2))=2^{2}-2=4-2=2$ .

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Example Question #2793 : Sat Mathematics

It takes no more than 40 minutes to run a race, but at least 30 minutes. What equation will model this in m minutes?

Possible Answers:

$\left | m-35 \right |= 5$ $\displaystyle \left | m-35 \right |= 5$

$\left | m-35 \right |< 5$ $\displaystyle \left | m-35 \right |< 5$

$\left | m+35 \right |> 5$ $\displaystyle \left | m+35 \right |> 5$

$\left | m+35 \right |< 5$ $\displaystyle \left | m+35 \right |< 5$

$\left | m-35 \right |> 5$ $\displaystyle \left | m-35 \right |> 5$

Correct answer:

$\left | m-35 \right |< 5$ $\displaystyle \left | m-35 \right |< 5$

Explanation:

If we take the mean number of minutes to be 35, then we need an equation which is less than 5 from either side of 35. If we subtract 35 from $m$ $\displaystyle m$ minutes and take the absolute value, this will give us our equation since we know that the time it takes to run the marathon is between 30 and 40 minutes.

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Example Question #91 : Algebraic Functions

If $\small f(x) = 4x^{2}+3x+2$ $\displaystyle \small f(x) = 4x^{2}+3x+2$ and $\small g(x) = x+7$ $\displaystyle \small g(x) = x+7$ , what is $\small f(g(x))$ $\displaystyle \small f(g(x))$ ?

Possible Answers:

$\small 4x^{2}+3x+219$ $\displaystyle \small 4x^{2}+3x+219$

$\small 4x^{2}+17x+219$ $\displaystyle \small 4x^{2}+17x+219$

$\small 4x^{2}+3x+72$ $\displaystyle \small 4x^{2}+3x+72$

$\small 4x^{2}+17x+72$ $\displaystyle \small 4x^{2}+17x+72$

$\small 4x^{2}+59x+219$ $\displaystyle \small 4x^{2}+59x+219$

Correct answer:

$\small 4x^{2}+59x+219$ $\displaystyle \small 4x^{2}+59x+219$

Explanation:

$\small 4(x+7)^{2} +3(x+7)+2$ $\displaystyle \small 4(x+7)^{2} +3(x+7)+2$

$\small 4(x^{2} + 14x + 49)+ 3x +21 + 2$ $\displaystyle \small 4(x^{2} + 14x + 49)+ 3x +21 + 2$

$\small 4x^{2}+56x+196+3x+23$ $\displaystyle \small 4x^{2}+56x+196+3x+23$

$\small 4x^{2}+59x+219$ $\displaystyle \small 4x^{2}+59x+219$

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Example Question #2791 : Sat Mathematics

If f(6)=7 $\displaystyle f(6)=7$ and $\displaystyle f(10)=17$ , which of the following could represent f(x) $\displaystyle f(x)$ ?

Possible Answers:

$\displaystyle 1.5x - 2$

2x + 5 $\displaystyle 2x + 5$

x + 4 $\displaystyle x + 4$

3x - 1 $\displaystyle 3x - 1$

$\displaystyle 2.5x - 8$

Correct answer:

$\displaystyle 2.5x - 8$

Explanation:

The number in the parentheses is what goes into the function.

For the function $\displaystyle f(x) = 2.5x - 8$ ,

$\displaystyle f(6) = 2.5(6) - 8 = 7$ and

$\displaystyle f(10) = 2.5(10) - 8 = 17$

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Example Question #2796 : Sat Mathematics

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)?

Possible Answers:

$\displaystyle 4$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\frac{7}{2}$ $\displaystyle \frac{7}{2}$

$\displaystyle 2$

$\displaystyle 0$

Correct answer:

$\displaystyle 2$

Explanation:

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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Example Question #421 : Gre Quantitative Reasoning

Which of the statements describes the solution set for –7(x + 3) = –7x + 20 ?

Possible Answers:

x = –1

All real numbers are solutions.

x = 0

There are no solutions.

Correct answer:

There are no solutions.

Explanation:

By distribution we obtain –7x – 21 = – 7x + 20. This equation is never possibly true.

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Example Question #92 : Algebraic Functions

Will just joined a poetry writing group in town that meets once a week. The number of poems Will has written after a certain number of meetings can be represented by the function $\displaystyle f(p) = 2p+6$ , where $\displaystyle p$ represents the number of meetings Will has attended. Using this function, how many poems has Will written after 7 classes?

Possible Answers:

$\displaystyle 20$

$\displaystyle 21$

$\displaystyle 15$

$\displaystyle 14$

$\displaystyle 19$

Correct answer:

$\displaystyle 20$

Explanation:

For this function, simply plug 7 in for $\displaystyle p$ and solve:

$\displaystyle f(p)=2p+6=2(7)+6=20$

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Example Question #101 : Algebraic Functions

If $f(x)=x^{2}+3$ $\displaystyle f(x)=x^{2}+3$ , then $f(x+h)=$ $\displaystyle f(x+h)=$ ?

Possible Answers:

$x^{2}+2xh+h^{2}+3$ $\displaystyle x^{2}+2xh+h^{2}+3$

$x^{2}+3+h$ $\displaystyle x^{2}+3+h$

$x^{2}+2xh+h^{2}$ $\displaystyle x^{2}+2xh+h^{2}$

$x^{2}+h^{2}$ $\displaystyle x^{2}+h^{2}$

$x^{2}+h^{2}+3$ $\displaystyle x^{2}+h^{2}+3$

Correct answer:

$x^{2}+2xh+h^{2}+3$ $\displaystyle x^{2}+2xh+h^{2}+3$

Explanation:

To find $f(x+h)$ $\displaystyle f(x+h)$ when $f(x)=x^{2}+3$ $\displaystyle f(x)=x^{2}+3$ , we substitute $(x+h)$ $\displaystyle (x+h)$ for $x$ $\displaystyle x$ in $f(x)$ $\displaystyle f(x)$ .

Thus, $f(x+h)=(x+h)^{2}+3$ $\displaystyle f(x+h)=(x+h)^{2}+3$ .

We expand $(x+h)^{2}$ $\displaystyle (x+h)^{2}$ to $x^{2}+xh+xh+h^{2}$ $\displaystyle x^{2}+xh+xh+h^{2}$ .

We can combine like terms to get $x^{2}+2xh+h^{2}$ $\displaystyle x^{2}+2xh+h^{2}$ .

We add 3 to this result to get our final answer.

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2791 : Sat Mathematics

Example Question #2792 : Sat Mathematics

Example Question #2792 : Sat Mathematics

Example Question #2793 : Sat Mathematics

Example Question #91 : Algebraic Functions

Example Question #2791 : Sat Mathematics

Example Question #2796 : Sat Mathematics

Example Question #421 : Gre Quantitative Reasoning

Example Question #92 : Algebraic Functions

Example Question #101 : Algebraic Functions

All ACT Math Resources

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