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Algebra 1 : How to find the domain of a function

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Example Questions

Example Question #1 : How To Find The Domain Of A Function

$\textup{What is the domain of the function }f(x)=\frac{x^{2}}{\sqrt{x+2}}?$

Possible Answers:

$x\geq-2$

x>2

x>-2

x=-2

Correct answer:

x>-2

Explanation:

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Example Question #1 : How To Find The Domain Of A Function

Define $f (x) = \sqrt[3]{\frac{100-x}{1,000-x}}$

What is the domain of ?

Possible Answers:

All real numbers except 100 and 1000

All real numbers except and

All real numbers except 1000

All real numbers except

All real numbers except , , 100 and 1000

Correct answer:

All real numbers except 1000

Explanation:

Every real number has a real cube root, so the radical does not restrict the domain of . The denominator of the expression does restrict the domain, however, in that it cannot be equal to 0. This happens only if:

1,000-x = 0 or, equivalently, x = 1,000 . Therefore, 1000 is the only real number not in the domain of .

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Example Question #1 : How To Find The Domain Of A Function

Find the domain of the following function:

$\sqrt[3]{x^{3} - 2x}$

Possible Answers:

$x^{2}> 2$

$x=\sqrt{2}$

$x^{2}\nleqslant 2$

$\left \{ x|-\infty< x< \infty \right \}$

$x\neq 2$

Correct answer:

$\left \{ x|-\infty< x< \infty \right \}$

Explanation:

The expression under the radical is defined for all real values of since the index of the radical is 3.

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Example Question #1 : How To Find The Domain Of A Function

Find the range of

$y=\sqrt{x-5}$

Possible Answers:

$\left \{ y\ |\ 0\leq y< \infty \right \}$

$\left \{ y\ |\ -\infty < y\leq 5 \right \}$

$\left \{ y\ |\ -\infty < y< \infty \right \}$

$\left \{ y\ |\ 0\leq y\leq 5 \right \}$

$\left ( x\ |\ 5\leq x< \infty \right )$

Correct answer:

$\left \{ y\ |\ 0\leq y< \infty \right \}$

Explanation:

Since the expression under the radical must be greater than or equal to zero, hence when x=5 , the f(x)=0 . Thereafter the f(x) is an increasing function.

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Example Question #2 : How To Find The Domain Of A Function

You are given a relation that comprises the following five points:

$\left \{(1,2), (3,N), (5,4), (N+3, 9), (2N-1, 10) \right \}$

For which value of is this relation a function?

Possible Answers:

N = 2

The relation is not a function for any of these values of .

N = - 2

N = 4

N = 0

Correct answer:

N = 4

Explanation:

A relation is a function if and only if no -coordinate is paired with more than one -coordinate. We test each of these four values of to see if this happens.

N = - 2 :

The points become:

(1,2), (3,-2), (5,4), (1, 9), (-5, 10)

Since -coordinate 1 is paired with two -coordinates, 2 and 9, the relation is not a function.

N = 0 :

The points become:

(1,2), (3,0), (5,4), (3, 9), (-1, 10)

Since -coordinate 3 is paired with two -coordinates, 0 and 9, the relation is not a function.

N = 2 :

The points become:

(1,2), (3,2), (5,4), (5, 9), (3, 10)

Since -coordinates 3 and 5 are each paired with two different -coordinates, the relation is not a function.

N = 4 :

The points become:

(1,2), (3,4), (5,4), (7, 9), (9, 10)

Since each -coordinate is paired with one and only one -coordinate, the relation is a function. N = 4 is the correct choice.

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Example Question #1 : How To Find The Domain Of A Function

Find the domain of the following function:

$f(x)=\sqrt{x+1}$

Possible Answers:

$x\leq 1$

$x\leq -1$

$x\geq -1$

$x\geq 1$

Domain can't be determined

Correct answer:

$x\geq -1$

Explanation:

Finding the domain is like finding out what possible values you can plug in without getting an error message on your calculator. We are given the function:

$f(x)=\sqrt{x+1}$

To find the domain, first, know that you can't take the square root of a negative number. It turns out that the minimum number you can take the square root of is zero. Any number above zero, everything is good. Therefore, x+1 can equal 0 or above. This is written as an inequality as:

$x+1\geq 0$

Solve for by subtracting 1 on both sides

$x+1-1\geq 0-1$

$x\geq -1$

So that's our domain!

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Example Question #1 : How To Find The Domain Of A Function

Give the domain of the function below.

$f(x) = \frac{1}{\sqrt{x^{2}+1}}$

Possible Answers:

$\left \{ x | x\neq -1\right \}$

$\mathbb{R}$

$\left \{ x | x\neq 1\right \}$

$\left \{x| -1<x<1 \right \}$

$\left \{ x | x< -1\; or \; x> 1\right \}$

Correct answer:

$\mathbb{R}$

Explanation:

The domain is the set of possible value for the variable. We can find the impossible values of by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.

$\sqrt{x^2+1}=0$

Now we can solve for .

x^2+1=0

x^2=-1

There is no real value of that will fit this equation; any real value squared will be a positive number.

The radicand is always positive, and $f(x) = \frac{1}{\sqrt{x^{2}+1}}$ is defined for all real values of . This makes the domain of the set of all real numbers.

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Example Question #2 : Domain And Range

What is the domain of the function $f(x)=\frac{2x}{x-2}$ ?

Possible Answers:

$\left ( -\infty ,-2 \right )\cup \left ( -2,\infty \right )$

$\left ( -\infty ,2 \right )\cup \left ( 2,\infty \right )$

$\left ( -\infty ,0 \right )\cup \left ( 0,2 \right )\cup \left ( 2,\infty \right )$

$\left ( -\infty ,\infty \right )$

$\left ( 2 \right )$

Correct answer:

$\left ( -\infty ,2 \right )\cup \left ( 2,\infty \right )$

Explanation:

The domain is the set of x-values that make the function defined.

This function is defined everywhere except at x = 2 , since division by zero is undefined.

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Example Question #1 : How To Find The Domain Of A Function

Give the domain of the function:

$f(x) = \frac{x^{2} - 16}{x^{2}-6x+9}$

Possible Answers:

$\left \{ x |x \neq -3 , x \neq 3 \right \}$

$\left \{ x |x \neq -3 \right \}$

$\left \{ x | x\neq -4, x \neq 3, x\neq 4\right \}$

$\left \{ x | x \neq 3 \right \}$

$\left \{ x |x \neq -4 , x \neq 4 \right \}$

Correct answer:

$\left \{ x | x \neq 3 \right \}$

Explanation:

The domain of a rational function is the set of all values of for which the denominator is not equal to 0, so we set the denominator to 0 and solve for .

$x^{2} -6x + 9 = 0$

This is a quadratic function, so we factor the expression as (x+?)(x+?) , replacing the question marks with two numbers whose product is 9 and whose sum is . These numbers are -3,-3 , so

$x^{2} -6x + 9 = 0$

becomes

(x-3)(x-3) = 0 ,

or $(x-3)^{2} = 0$ .

This means that x-3=0 , or x = 3 .

Therefore, 3 is the only number excluded from the domain.

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Example Question #4 : How To Find The Domain Of A Function

Give the domain of the function:

$f(x) = \frac{x^{2} - 4}{x^{2}-6x+8}$

Possible Answers:

$\left \{x | x \neq -2, x \neq 2\ \right \}$

$\left \{x | x \neq -2, x \neq 2 ,x \neq 4 \right \}$

The set of all real numbers.

$\left \{x | x \neq 2 ,x \neq 4 \right \}$

$\left \{x | x \neq 2 \right \}$

Correct answer:

$\left \{x | x \neq 2 ,x \neq 4 \right \}$

Explanation:

The domain of a rational function is the set of all values of for which the denominator is not equal to 0 (the value of the numerator is irrelevant), so we set the denominator to 0 and solve for to find the excluded values.

$x^{2}-6x+8 =0$

This is a quadratic function, so we factor the expression as (x+?)(x+?) , replacing the question marks with two numbers whose product is 8 and whose sum is . These numbers are -4,-2 , so

$x^{2}-6x+8 =0$

becomes

(x-2)(x-4) = 0

So either

x-2 = 0 , in which case x = 2 ,

or x-4 = 0 , in which case x = 4 .

Therefore, 2 and 4 are the only numbers excluded from the domain.

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Algebra 1 : How to find the domain of a function

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Find The Domain Of A Function

Example Question #1 : How To Find The Domain Of A Function

Example Question #1 : How To Find The Domain Of A Function

Example Question #1 : How To Find The Domain Of A Function

Example Question #2 : How To Find The Domain Of A Function

Example Question #1 : How To Find The Domain Of A Function

Example Question #1 : How To Find The Domain Of A Function

Example Question #2 : Domain And Range

Example Question #1 : How To Find The Domain Of A Function

Example Question #4 : How To Find The Domain Of A Function

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