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Precalculus : Algebra of Functions

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Example Questions

Example Question #5 : Add, Subtract, Multiply, And Divide Functions

Simplify the following expression: $(6x^{3}-3x^{2}+4x+9)-(6x^{3}-5x^{2}+7x+8)$

Possible Answers:

$2x^{2}-3x+1$

No correct answer listed

$8x^{2}-5x+1$

$6x^{3}-2x^{2}+4x+17$

$8x^{2}-3x+1$

Correct answer:

$2x^{2}-3x+1$

Explanation:

To simplify the above expression, we must combine all like terms:

$(6x^{3}-3x^{2}+4x+9)-(6x^{3}-5x^{2}+7x+8)$

$x^{3}$ : $6x^{3}-6x^{3}=0$

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

: 4x-7x=-3x

Integers: 9-8=1

Putting all of the above terms together, we simplify to:

$2x^{2}-3x+1$

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Example Question #4 : Add, Subtract, Multiply, And Divide Functions

If $f(x)=5x^{2}-15$ and $g(x)=12x^{2}-36$ , what is $\frac{f(x)}{g(x)}$ ?

Possible Answers:

$\frac{3}{5}$

$\frac{5}{12}$

$\frac{1}{3}$

$\frac{1}{2}$

$\frac{7}{12}$

Correct answer:

$\frac{5}{12}$

Explanation:

Given the information in the above problem, we know that:

$\frac{f(x)}{g(x)}=\frac{5x^{2}-15}{12x^{2}-36}$

Factoring the resulting fraction, we get:

$\frac{5x^{2}-15}{12x^{2}-36}$

$=\frac{5(x^{2}-3)}{12(x^{2}-3)}$

$=\frac{5}{12}$

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Example Question #10 : Add, Subtract, Multiply, And Divide Functions

Simplify the following:

(x^2-4x+3)-(3x^2+2x-6)

Possible Answers:

-2x^2-6x+9

-2x^2-2x-3

4x^2-6x+9

4x^2-2x-3

Correct answer:

-2x^2-6x+9

Explanation:

To simplify the expression, distribute the negative into the second parentheses, and then combine like terms.

(x^2-4x+3)-(3x^2+2x-6)

x^2-4x+3-3x^2-2x+6

x^2-3x^2-4x-2x+3+6

-2x^2-6x+9

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Example Question #11 : Algebra Of Functions

Simplify the following completely:

(3x^2+2x+1)+(2x+1)

Possible Answers:

5x^2+4x+2

3x^2+2x+2x+1+1

3x^2+4x+2

5x^2+3x+1

Correct answer:

3x^2+4x+2

Explanation:

To simlify adding polynomials, simply drop the parentheses and add like terms.

3x^2+2x+1+2x+1

3x^2+4x+2

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Example Question #12 : Add, Subtract, Multiply, And Divide Functions

Determine the sum of: $\frac{a+b}{c}+ \frac{c}{a}$

Possible Answers:

$\frac{a+b+c}{a+c}$

$\frac{2a+ab+2c}{ac}$

$\frac{a^2+ab+c^2}{a+c}$

$\frac{a^2+ab+c^2}{ac}$

$\frac{a+b+c}{ac}$

Correct answer:

$\frac{a^2+ab+c^2}{ac}$

Explanation:

To add the numerators, the denominators must be common.

The least common denominator can be determined by multiplication.

$a\times c= ac$

Rewrite the fractions.

$\frac{a+b}{c}+ \frac{c}{a} = \frac{a(a+b)}{ac}+\frac{c(c)}{ac} = \frac{a^2+ab+c^2}{ac}$

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Example Question #1101 : Pre Calculus

Given f(x) and g(x) ,

f(x)=x^2-3x+7

g(x)=2x^3+5x^2-26

Complete the operation given by g(x)-f(x) .

Possible Answers:

2x^3+4x^2+3x-33

2x^3+4x^2-33

2x^3+4x^2+3x-43

6x^3+3x-33

Correct answer:

2x^3+4x^2+3x-33

Explanation:

Given f(x) and g(x)

f(x)=x^2-3x+7

g(x)=2x^3+5x^2-26

Complete the operation given by g(x)-f(x) .

Begin by realizing what this is asking. We need to combine our two functions in such a way that we find the difference between them.

When doing so remember to distribute the negative sign that is in front of f(x) to each term within the polynomial.

$\\(2x^3+5x^2-26)-(x^2-3x+7)\\ \\=2x^3+5x^2-26-x^2+3x-7\\ \\=2x^3+4x^2+3x-33$

So, by simplifying the expression, we get our answer to be:

2x^3+4x^2+3x-33

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Example Question #1102 : Pre Calculus

Given f(x) and g(x) ,

f(x)=x^2-3x+7

g(x)=2x^3+5x^2-26

Evaluate and simplify .

Possible Answers:

4x^3-11x^2-45

4x^3+11x^2-3x-59

4x^3+11x^2-3x-45

4x^3+9x^2-3x-45

Correct answer:

4x^3+11x^2-3x-45

Explanation:

Given f(x) and g(x) ,

f(x)=x^2-3x+7

g(x)=2x^3+5x^2-26

Evaluate and simplify .

Begin by multiplying g(x) by 2:

2g(x)=2(2x^3+5x^2-26)=4x^3+10x^2-52

Next, add f(x) to what we got above and combine like terms.

$\\4x^3+10x^2-52+(x^2-3x+7)\\ \\=4x^3=10x^2+x^2-3x-52+7\\ \\=4x^3+11x^2-3x-45$

This makes our answer

4x^3+11x^2-3x-45 .

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Example Question #1103 : Pre Calculus

Given y(t) and b(t) , find y(t)*b(t) .

y(t)=t^3-8t

b(t)=5t^7+8t^2

Possible Answers:

$5t^{21}-40t^{15}+8t^6-64t^2$

$5t^{10}-40t^8+8t^5$

$5t^{10}-40t^8+8t^5-64t^3$

$15t^{10}-45t^8+18t^5-16t^3$

Correct answer:

$5t^{10}-40t^8+8t^5-64t^3$

Explanation:

Given y(t) and b(t) , find y(t)*b(t) .

y(t)=t^3-8t

b(t)=5t^7+8t^2

To complete this problem, we need to recall FOIL. FOIL states to multiply the terms in each binomial together in the order of first, outer, inner, and last.

$y(t)*b(t)=(t^3-8t) (5t^7+8t^2)={\color{Blue} t^3*5t^7}+{\color{Red} t^3*8t^2}+{\color{Magenta} -8t*5t^7}+{\color{DarkOrange} -8t*8t^2}$

${\color{Blue} t^3*5t^7}+{\color{Red} t^3*8t^2}+{\color{Magenta} -8t*5t^7}+{\color{DarkOrange} -8t*8t^2}=5t^{10}+8t^5-40t^8-64t^3$

We have no like terms to combine, so our answer is:

$5t^{10}-40t^8+8t^5-64t^3$

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Example Question #12 : Algebra Of Functions

Determine $\small f+g$

$\small f(x)=3x$ and $\small g(x)=7$

Possible Answers:

$\small f+g=3x+7$

$\small f+g=4x$

$\small f+g=10$

$\small f+g=10x$

Correct answer:

$\small f+g=3x+7$

Explanation:

f+g is defined as the sum of the two functions f(x) and $\small g(x)$ .

As such

$\small f+g=f(x)+g(x)=3x+7$

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Example Question #12 : Algebra Of Functions

Determine $\small f+g$

$\small \small f(x)=5x^2$ and $\small \small g(x)=6x^2$

Possible Answers:

$\small \small f+g = 11x^2$

$\small \small \small f+g = \frac{5}{6}$

$\small \small \small f+g = 30x^4$

$\small \small \small f+g = x^2$

Correct answer:

$\small \small f+g = 11x^2$

Explanation:

f+g is defined as the sum of the two functions f(x) and $\small g(x)$ .

As such

$\small \small f+g=f(x)+g(x)=5x^2+6x^2=11x^2$

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Precalculus : Algebra of Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #5 : Add, Subtract, Multiply, And Divide Functions

Example Question #4 : Add, Subtract, Multiply, And Divide Functions

Example Question #10 : Add, Subtract, Multiply, And Divide Functions

Example Question #11 : Algebra Of Functions

Example Question #12 : Add, Subtract, Multiply, And Divide Functions

Example Question #1101 : Pre Calculus

Example Question #1102 : Pre Calculus

Example Question #1103 : Pre Calculus

Example Question #12 : Algebra Of Functions

Example Question #12 : Algebra Of Functions

All Precalculus Resources

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