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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #62 : Introduction To Functions

What is the domain of the function $f(x)=\sqrt{x-2}+3$ $\displaystyle f(x)=\sqrt{x-2}+3$?

Possible Answers:

$[3,\infty)$ $\displaystyle [3,\infty)$

$(-\infty, 2]$ $\displaystyle (-\infty, 2]$

$[-2,\infty )$ $\displaystyle [-2,\infty )$

$[-3,\infty)$ $\displaystyle [-3,\infty)$

$[2,\infty)$ $\displaystyle [2,\infty)$

Correct answer:

$[2,\infty)$ $\displaystyle [2,\infty)$

Explanation:

The expression under the square root symbol cannot be negative, so to find the domain, set that expression $\geq0$ $\displaystyle \geq0$.

$x-2\geq0$ $\displaystyle x-2\geq0$

$x\geq2$ $\displaystyle x\geq2$

The domain includes all x-values greater than or equal to 2, which can be written as $[2,\infty)$ $\displaystyle [2,\infty)$.

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Example Question #62 : Introduction To Functions

What is the domain of the function $f(x)=\sqrt{7-x}+2$ $\displaystyle f(x)=\sqrt{7-x}+2$?

Possible Answers:

$(-\infty, 7]$ $\displaystyle (-\infty, 7]$

$[-7,\infty)$ $\displaystyle [-7,\infty)$ $[3,\infty)$ $\displaystyle [3,\infty)$

$(-\infty, 2]$ $\displaystyle (-\infty, 2]$

$[2,\infty)$ $\displaystyle [2,\infty)$

$[7,\infty)$ $\displaystyle [7,\infty)$

Correct answer:

$(-\infty, 7]$ $\displaystyle (-\infty, 7]$

Explanation:

The expression under the square root symbol cannot be negative, so to find the domain, set that expression $\geq0$ $\displaystyle \geq0$.

$7-x\geq0$ $\displaystyle 7-x\geq0$

$-x\geq-2$ $\displaystyle -x\geq-2$

$x\leq7$ $\displaystyle x\leq7$

The domain includes all x-values less than or equal to 7, which can be written as $(-\infty,7]$ $\displaystyle (-\infty,7]$.

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Example Question #591 : Algebra Ii

What is the range of the function $f(x)=\sqrt{x+4}-6$ $\displaystyle f(x)=\sqrt{x+4}-6$?

Possible Answers:

$[-6,\infty)$ $\displaystyle [-6,\infty)$

$(-\infty,-6]$ $\displaystyle (-\infty,-6]$

$[-4,\infty)$ $\displaystyle [-4,\infty)$

$[-6,\infty)$ $\displaystyle [-6,\infty)$

$[4,\infty)$ $\displaystyle [4,\infty)$

Correct answer:

$[-6,\infty)$ $\displaystyle [-6,\infty)$

Explanation:

The smallest value that the function can have is when the square root part of the function is zero (it cannot be smaller than zero).

0-6=-6 $\displaystyle 0-6=-6$

The smallest value is -6, so the range must be $[-6,\infty)$ $\displaystyle [-6,\infty)$

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Example Question #42 : Domain And Range

What is the range of the function $f(x)=-\sqrt{x+7}+3$ $\displaystyle f(x)=-\sqrt{x+7}+3$?

Possible Answers:

$[3,\infty)$ $\displaystyle [3,\infty)$

$[-7,\infty)$ $\displaystyle [-7,\infty)$

$(-\infty,-3]$ $\displaystyle (-\infty,-3]$

$(-\infty,3]$ $\displaystyle (-\infty,3]$

$(-\infty,-7]$ $\displaystyle (-\infty,-7]$

Correct answer:

$(-\infty,3]$ $\displaystyle (-\infty,3]$

Explanation:

The largest value that the function can have is when the square root part of the function is zero (it cannot be smaller than zero).

-0+3=3 $\displaystyle -0+3=3$

The largest value is 3, so the range must be $(-\infty,3]$ $\displaystyle (-\infty,3]$

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Example Question #42 : Domain And Range

What is the range of the function $f(x)=-\sqrt{x+7}+3$ $\displaystyle f(x)=-\sqrt{x+7}+3$?

Possible Answers:

$[3,\infty)$ $\displaystyle [3,\infty)$

$[-7,\infty)$ $\displaystyle [-7,\infty)$

$(-\infty,-3]$ $\displaystyle (-\infty,-3]$

$(-\infty,3]$ $\displaystyle (-\infty,3]$

$(-\infty,-7]$ $\displaystyle (-\infty,-7]$

Correct answer:

$(-\infty,3]$ $\displaystyle (-\infty,3]$

Explanation:

The largest value that the function can have is when the square root part of the function is zero (it cannot be smaller than zero).

-0+3=3 $\displaystyle -0+3=3$

The largest value is 3, so the range must be $(-\infty,3]$ $\displaystyle (-\infty,3]$

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Example Question #43 : Domain And Range

What is the domain and range of the following equation:

$\small y=x^2+5$ $\displaystyle \small y=x^2+5$

Possible Answers:

$\small Domain: (-\infty, \infty)$ $\displaystyle \small Domain: (-\infty, \infty)$

$\small Range: [5, \infty)$ $\displaystyle \small Range: [5, \infty)$

$\small Domain: (5, \infty)$ $\displaystyle \small Domain: (5, \infty)$

$\small Range: (-\infty, \infty)$ $\displaystyle \small Range: (-\infty, \infty)$

$\small Domain: [5, \infty)$ $\displaystyle \small Domain: [5, \infty)$

$\small Range: (-\infty, \infty)$ $\displaystyle \small Range: (-\infty, \infty)$

$\small Domain: [-\infty, \infty]$ $\displaystyle \small Domain: [-\infty, \infty]$

$\small Range: [5, \infty)$ $\displaystyle \small Range: [5, \infty)$

$\small Domain: (-\infty, \infty)$ $\displaystyle \small Domain: (-\infty, \infty)$

$\small Range: (5, \infty)$ $\displaystyle \small Range: (5, \infty)$

Correct answer:

$\small Domain: (-\infty, \infty)$ $\displaystyle \small Domain: (-\infty, \infty)$

$\small Range: [5, \infty)$ $\displaystyle \small Range: [5, \infty)$

Explanation:

The domain of any quadratic function is always all real numbers.

The range of this function is anything greater than or equal to 5.

These written in the correct notation is:

$\small Domain: (-\infty, \infty)$ $\displaystyle \small Domain: (-\infty, \infty)$

$\small Range: [5, \infty)$ $\displaystyle \small Range: [5, \infty)$

Soft brackets are needed for infinity and a hard square bracket for 5 because it is included in the solution.

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Example Question #43 : Domain And Range

What is the domain of the following function?

$\small \frac{5}{x+5}$ $\displaystyle \small \frac{5}{x+5}$

Possible Answers:

$\small (-\infty -5]\cup(-5, \infty)$ $\displaystyle \small (-\infty -5]\cup(-5, \infty)$

$\small (-\infty -5]\cup[-5, \infty)$ $\displaystyle \small (-\infty -5]\cup[-5, \infty)$

$\small [-\infty -5)\cup(-5, \infty]$ $\displaystyle \small [-\infty -5)\cup(-5, \infty]$

$\small (-\infty -5)\cup(-5, \infty)$ $\displaystyle \small (-\infty -5)\cup(-5, \infty)$

$\small [-\infty -5]\cup[-5, \infty]$ $\displaystyle \small [-\infty -5]\cup[-5, \infty]$

Correct answer:

$\small (-\infty -5)\cup(-5, \infty)$ $\displaystyle \small (-\infty -5)\cup(-5, \infty)$

Explanation:

To figure out the values that x (your domain) can't be you need to set the denominator equal to 0 since you cannot have 0 in the denominator.

When you solve this you get $\small x=-5$ $\displaystyle \small x=-5$

This means that you have have all values of x other than -5.

This notation of this is:

$\small (-\infty -5)\cup(-5, \infty)$ $\displaystyle \small (-\infty -5)\cup(-5, \infty)$

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Example Question #42 : Domain And Range

What is the domain and range of the graph below?

1 over x

Possible Answers:

$\small Domain: (-\infty, 0]\cup[0, \infty)$ $\displaystyle \small Domain: (-\infty, 0]\cup[0, \infty)$

$\small Range: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Range: (-\infty, 0)\cup(0, \infty)$

$\small Domain: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Domain: (-\infty, 0)\cup(0, \infty)$

$\small Range: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Range: (-\infty, 0)\cup(0, \infty)$

$\small Domain: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Domain: (-\infty, 0)\cup(0, \infty)$

$\small Range: (-\infty, 0]\cup[0, \infty)$ $\displaystyle \small Range: (-\infty, 0]\cup[0, \infty)$

$\small Domain: (-\infty, \infty)$ $\displaystyle \small Domain: (-\infty, \infty)$

$\small Range: (-\infty, \infty)$ $\displaystyle \small Range: (-\infty, \infty)$

None of the above

Correct answer:

$\small Domain: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Domain: (-\infty, 0)\cup(0, \infty)$

$\small Range: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Range: (-\infty, 0)\cup(0, \infty)$

Explanation:

This function has the domain of x can't be 0 as well as the range of y can't be 0.

This means that the domain and range of these functions include everything but 0, which is denoted as:

$\small Domain: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Domain: (-\infty, 0)\cup(0, \infty)$ and $\small Range: (-\infty, 0)\cup(0, \infty)$ $\displaystyle \small Range: (-\infty, 0)\cup(0, \infty)$

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Example Question #592 : Algebra Ii

What is the range of the following inequality:

$y+2\geq20$ $\displaystyle y+2\geq20$

Possible Answers:

$(18,\infty)$ $\displaystyle (18,\infty)$

$(-\infty,18)$ $\displaystyle (-\infty,18)$

$[18, \infty]$ $\displaystyle [18, \infty]$

$(-\infty,18]$ $\displaystyle (-\infty,18]$

$[18,\infty)$ $\displaystyle [18,\infty)$

Correct answer:

$[18,\infty)$ $\displaystyle [18,\infty)$

Explanation:

Upon solving this inequality you get

$y\geq18$ $\displaystyle y\geq18$

This means that "y" is all values including 18 and greater.

Written as interval notation you get:

$[18, \infty)$ $\displaystyle [18, \infty)$

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Example Question #41 : Domain And Range

What is the domain of the following equation:

$y=\frac{x^2+4x+3}{x+3}$ $\displaystyle y=\frac{x^2+4x+3}{x+3}$

Possible Answers:

$(-\infty, \infty)$ $\displaystyle (-\infty, \infty)$

$(-\infty, 3)\cup (3, \infty)$ $\displaystyle (-\infty, 3)\cup (3, \infty)$

$(-\infty, -3)\cup (-3, \infty)$ $\displaystyle (-\infty, -3)\cup (-3, \infty)$

$(-\infty, 3]\cup [3, \infty)$ $\displaystyle (-\infty, 3]\cup [3, \infty)$

$(-\infty, -3]\cup [-3, \infty)$ $\displaystyle (-\infty, -3]\cup [-3, \infty)$

Correct answer:

$(-\infty, \infty)$ $\displaystyle (-\infty, \infty)$

Explanation:

The trick here is to factor the top. Upon doing so you get:

$y=\frac{(x+3)(x+1)}{x+3}$ $\displaystyle y=\frac{(x+3)(x+1)}{x+3}$

From here you can cancel your "x+3"s to get

y=x+1 $\displaystyle y=x+1$ which has a domain of $(-\infty, \infty)$ $\displaystyle (-\infty, \infty)$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #62 : Introduction To Functions

Example Question #62 : Introduction To Functions

Example Question #591 : Algebra Ii

Example Question #42 : Domain And Range

Example Question #42 : Domain And Range

Example Question #43 : Domain And Range

Example Question #43 : Domain And Range

Example Question #42 : Domain And Range

Example Question #592 : Algebra Ii

Example Question #41 : Domain And Range

All Algebra II Resources

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