Use Radicals in Functions

Evaluate a Radical Function

In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a radical function.

The square root function is \(f(x)=\sqrt[]{x}.\)

The cube root function is \(f(x)=\sqrt[3]{x}.\)

To evaluate a radical function, we find the value of f(x) for a given value of x just as we did in our previous work with functions.

Example

Try it.

For the function \(f(x)=\sqrt{2x-1},\) find ⓐ \(f(5)\) ⓑ \(f(-2).\)

Solution

ⓐ

	\(f(x)=\sqrt{2x-1}\)
To evaluate \(f(5),\) substitute 5 for \(x.\)	\(f(5)=\sqrt{2\cdot 5-1}\)
Simplify.	\(f(5)=\sqrt{9}\)
Take the square root.	\(f(5)=3\)

ⓑ

	\(\ f(x)=\sqrt{2x-1}\)
To evaluate \(f(-2),\) substitute −2 for \(x.\)	\(\ f(-2)=\sqrt{2(-2)-1}\)
Simplify.	\(\ f(-2)=\sqrt{-5}\)

Since the square root of a negative number is not a real number, the function does not have a value at \(x=-2.\)

We follow the same procedure to evaluate cube roots.

Example

Try it.

For the function \(g(x)=\sqrt[3]{x-6},\) find ⓐ \(g(14)\) ⓑ \(g(-2).\)

Solution

ⓐ

	\(\ g(x)=\sqrt[3]{x-6}\)
To evaluate \(g(14),\) substitute 14 for \(x.\)	\(\ g(14)=\sqrt[3]{14-6}\)
Simplify.	\(\ g(14)=\sqrt[3]{8}\)
Take the cube root.	\(\ g(14)=2\)

ⓑ

	\(\ g(x)=\sqrt[3]{x-6}\)
To evaluate \(g(-2),\) substitute −2 for \(x.\)	\(\ g(-2)=\sqrt[3]{-2-6}\)
Simplify.	\(\ g(-2)=\sqrt[3]{-8}\)
Take the cube root.	\(\ g(-2)=-2\)

The next example has fourth roots.

Example

Try it.

For the function \(f(x)=\sqrt[4]{5x-4},\) find ⓐ \(f(4)\) ⓑ \(f(-12)\)

Solution

ⓐ

	\(\ f(x)=\sqrt[4]{5x-4}\)
To evaluate \(f(4),\) substitute 4 for \(x.\)	\(\ f(4)=\sqrt[4]{5\cdot 4-4}\)
Simplify.	\(\ f(4)=\sqrt[4]{16}\)
Take the fourth root.	\(\ f(4)=2\)

ⓑ

	\(\ f(x)=\sqrt[4]{5x-4}\)
To evaluate \(f(-12),\) substitute −12 for \(x.\)	\(\ f(-12)=\sqrt[4]{5(-12)-4}\)
Simplify.	\(\ f(-12)=\sqrt[4]{-64}\)

Since the fourth root of a negative number is not a real number, the function does not have a value at \(x=-12.\)

Find the Domain of a Radical Function

To find the domain and range of radical functions, we use our properties of radicals. For a radical with an even index, we said the radicand had to be greater than or equal to zero as even roots of negative numbers are not real numbers. For an odd index, the radicand can be any real number. We restate the properties here for reference.

So, to find the domain of a radical function with even index, we set the radicand to be greater than or equal to zero. For an odd index radical, the radicand can be any real number.

Example

Try it.

Find the domain of the function, \(f(x)=\sqrt{3x-4}.\) Write the domain in interval notation.

Solution

Since the function, \(f(x)=\sqrt{3x-4}\) has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0. We set the radicand to be greater than or equal to 0 and then solve to find the domain.

	\(3x-4\ge 0\)
Solve.	\(3x\ge 4\)
	\(x\ge \frac{4}{3}\)

The domain of \(f(x)=\sqrt{3x-4}\) is all values \(x\ge \frac{4}{3}\) and we write it in interval notation as \([\frac{4}{3},\infty ).\)

Example

Try it.

Find the domain of the function, \(g(x)=\sqrt{\frac{6}{x-1}}.\) Write the domain in interval notation.

Solution

Since the function, \(g(x)=\sqrt{\frac{6}{x-1}}\) has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0.

The radicand cannot be zero since the numerator is not zero.

For \(\frac{6}{x-1}\) to be greater than zero, the denominator must be positive since the numerator is positive. We know a positive divided by a positive is positive.

We set \(x-1>0\) and solve.

Solve.

\(\begin{array}{lll}x-1 & > & 0 \\ x & > & 1\end{array}\)

Also, since the radicand is a fraction, we must realize that the denominator cannot be zero.

We solve \(x-1=0\) to find the value that must be eliminated from the domain.

Solve.

\(\begin{array}{lll}x-1 & = & 0 \\ x & = & 1\ \text{so}\ x\ne 1\ \text{in the domain.}\end{array}\)

Putting this together we get the domain is \(x>1\) and we write it as \((1,\infty ).\)

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Graph Radical Functions

Before we graph any radical function, we first find the domain of the function. For the function, \(f(x)=\sqrt{x},\) the index is even, and so the radicand must be greater than or equal to 0.

This tells us the domain is \(x\ge 0\) and we write this in interval notation as \([0,\infty ).\)

Previously we used point plotting to graph the function, \(f(x)=\sqrt{x}.\) We chose x-values, substituted them in and then created a chart. Notice we chose points that are perfect squares in order to make taking the square root easier.

Once we see the graph, we can find the range of the function. The y-values of the function are greater than or equal to zero. The range then is \([0,\infty ).\)

Example

Try it.

For the function \(f(x)=\sqrt{x+3},\)

ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

Solution

ⓐ Since the radical has index 2, we know the radicand must be greater than or equal to zero. If \(x+3\ge 0,\) then \(x\ge -3.\) This tells us the domain is all values \(x\ge -3\) and written in interval notation as \([-3,\infty ).\)

ⓑ To graph the function, we choose points in the interval \([-3,\infty )\) that will also give us a radicand which will be easy to take the square root.

ⓒ Looking at the graph, we see the y-values of the function are greater than or equal to zero. The range then is \([0,\infty ).\)

In our previous work graphing functions, we graphed \(f(x)={x}^{3}\) but we did not graph the function \(f(x)=\sqrt[3]{x}.\) We will do this now in the next example.

Example

Try it.

For the function \(f(x)=\sqrt[3]{x},\) ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

Solution

ⓐ Since the radical has index 3, we know the radicand can be any real number. This tells us the domain is all real numbers and written in interval notation as \((\text{-}\infty ,\infty )\)

ⓑ To graph the function, we choose points in the interval \((\text{-}\infty ,\infty )\) that will also give us a radicand which will be easy to take the cube root.

ⓒ Looking at the graph, we see the y-values of the function are all real numbers. The range then is \((\text{-}\infty ,\infty ).\)

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• Properties of \(\sqrt[n]{a}\)
  • When n is an even number and:
    \(a\ge 0,\) then \(\sqrt[n]{a}\) is a real number.
    \(a<0,\) then \(\sqrt[n]{a}\) is not a real number.
  • When n is an odd number, \(\sqrt[n]{a}\) is a real number for all values of a.
• Domain of a Radical Function
  • When the index of the radical is even, the radicand must be greater than or equal to zero.
  • When the index of the radical is odd, the radicand can be any real number.

Use Radicals in Functions

Evaluate a Radical Function

In the following exercises, evaluate each function.

Try it.

\(f(x)=\sqrt{4x-4},\) find ⓐ \(f(5)\) ⓑ \(f(0).\)

Solution

ⓐ \(f(5)=4\) ⓑ no value at \(x=0\)

Try it.

\(f(x)=\sqrt{6x-5},\) find ⓐ \(f(5)\) ⓑ \(f(-1).\)

Try it.

\(g(x)=\sqrt{6x+1},\) find ⓐ \(g(4)\) ⓑ \(g(8).\)

Solution

ⓐ \(g(4)=5\) ⓑ \(g(8)=7\)

Try it.

\(g(x)=\sqrt{3x+1},\) find ⓐ \(g(8)\) ⓑ \(g(5).\)

Try it.

\(F(x)=\sqrt{3-2x},\) find ⓐ \(F(1)\) ⓑ \(F(-11).\)

Solution

ⓐ \(F(1)=1\) ⓑ \(F(-11)=5\)

Try it.

\(F(x)=\sqrt{8-4x},\) find ⓐ \(F(1)\) ⓑ \(F(-2).\)

Try it.

\(G(x)=\sqrt{5x-1},\) find ⓐ \(G(5)\) ⓑ \(G(2).\)

Solution

ⓐ \(G(5)=2\sqrt{6}\) ⓑ \(G(2)=3\)

Try it.

\(G(x)=\sqrt{4x+1},\) find ⓐ \(G(11)\) ⓑ \(G(2).\)

Try it.

\(g(x)=\sqrt[3]{2x-4},\) find ⓐ \(g(6)\) ⓑ \(g(-2).\)

Solution

ⓐ \(g(6)=2\) ⓑ \(g(-2)=-2\)

Try it.

\(g(x)=\sqrt[3]{7x-1},\) find ⓐ \(g(4)\) ⓑ \(g(-1).\)

Try it.

\(h(x)=\sqrt[3]{{x}^{2}-4},\) find ⓐ \(h(-2)\) ⓑ \(h(6).\)

Solution

ⓐ \(h(-2)=0\) ⓑ \(h(6)=2\sqrt[3]{4}\)

Try it.

\(h(x)=\sqrt[3]{{x}^{2}+4},\) find ⓐ \(h(-2)\) ⓑ \(h(6).\)

Try it.

For the function \(f(x)=\sqrt[4]{2{x}^{3}},\) find ⓐ \(f(0)\) ⓑ \(f(2).\)

Solution

ⓐ \(f(0)=0\) ⓑ \(f(2)=2\)

Try it.

For the function \(f(x)=\sqrt[4]{3{x}^{3}},\) find ⓐ \(f(0)\) ⓑ \(f(3).\)

Try it.

For the function \(g(x)=\sqrt[4]{4-4x},\) find ⓐ \(g(1)\) ⓑ \(g(-3).\)

Solution

ⓐ \(g(1)=0\) ⓑ \(g(-3)=2\)

Try it.

For the function \(g(x)=\sqrt[4]{8-4x},\) find ⓐ \(g(-6)\) ⓑ \(g(2).\)

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

Try it.

\(f(x)=\sqrt{3x-1}\)

Solution

\([\frac{1}{3},\infty )\)

Try it.

\(f(x)=\sqrt{4x-2}\)

Try it.

\(g(x)=\sqrt{2-3x}\)

Solution

\((\text{-}\infty ,\frac{2}{3}]\)

Try it.

\(g(x)=\sqrt{8-x}\)

Try it.

\(h(x)=\sqrt{\frac{5}{x-2}}\)

Solution

\((2,\infty )\)

Try it.

\(h(x)=\sqrt{\frac{6}{x+3}}\)

Try it.

\(f(x)=\sqrt{\frac{x+3}{x-2}}\)

Solution

\((\text{-}\infty ,-3]\cup (2,\infty )\)

Try it.

\(f(x)=\sqrt{\frac{x-1}{x+4}}\)

Try it.

\(g(x)=\sqrt[3]{8x-1}\)

Solution

\((\text{-}\infty ,\infty )\)

Try it.

\(g(x)=\sqrt[3]{6x+5}\)

Try it.

\(f(x)=\sqrt[3]{4{x}^{2}-16}\)

Solution

\((\text{-}\infty ,\infty )\)

Try it.

\(f(x)=\sqrt[3]{6{x}^{2}-25}\)

Try it.

\(F(x)=\sqrt[4]{8x+3}\)

Solution

\([-\frac{3}{8},\infty )\)

Try it.

\(F(x)=\sqrt[4]{10-7x}\)

Try it.

\(G(x)=\sqrt[5]{2x-1}\)

Solution

\((\text{-}\infty ,\infty )\)

Try it.

\(G(x)=\sqrt[5]{6x-3}\)

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

Try it.

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

Solution

ⓐ domain: \([-1,\infty )\)
ⓑ

ⓒ \([0,\infty )\)

Try it.

\(f(x)=\sqrt{x-1}\)

Try it.

\(g(x)=\sqrt{x+4}\)

Solution

ⓐ domain: \([-4,\infty )\)
ⓑ

ⓒ \([0,\infty )\)

Try it.

\(g(x)=\sqrt{x-4}\)

Try it.

\(f(x)=\sqrt{x}+2\)

Solution

ⓐ domain: \([0,\infty )\)
ⓑ

ⓒ \([2,\infty )\)

Try it.

\(f(x)=\sqrt{x}-2\)

Try it.

\(g(x)=2\sqrt{x}\)

Solution

ⓐ domain: \([0,\infty )\)
ⓑ

ⓒ \([0,\infty )\)

Try it.

\(g(x)=3\sqrt{x}\)

Try it.

\(f(x)=\sqrt{3-x}\)

Solution

ⓐ domain: \((\text{-}\infty ,3]\)
ⓑ

ⓒ \([0,\infty )\)

Try it.

\(f(x)=\sqrt{4-x}\)

Try it.

\(g(x)=\text{-}\sqrt{x}\)

Solution

ⓐ domain: \([0,\infty )\)
ⓑ

ⓒ \((\text{-}\infty ,0]\)

Try it.

\(g(x)=\text{-}\sqrt{x}+1\)

Try it.

\(f(x)=\sqrt[3]{x+1}\)

Solution

ⓐ domain: \((\text{-}\infty ,\infty )\)
ⓑ

ⓒ \((\text{-}\infty ,\infty )\)

Try it.

\(f(x)=\sqrt[3]{x-1}\)

Try it.

\(g(x)=\sqrt[3]{x+2}\)

Solution

ⓐ domain: \((\text{-}\infty ,\infty )\)
ⓑ

ⓒ \((\text{-}\infty ,\infty )\)

Try it.

\(g(x)=\sqrt[3]{x-2}\)

Try it.

\(f(x)=\sqrt[3]{x}+3\)

Solution

ⓐ domain: \((\text{-}\infty ,\infty )\)
ⓑ

ⓒ \((\text{-}\infty ,\infty )\)

Try it.

\(f(x)=\sqrt[3]{x}-3\)

Try it.

\(g(x)=\sqrt[3]{x}\)

Solution

ⓐ domain: \((\text{-}\infty ,\infty )\)
ⓑ

ⓒ \((\text{-}\infty ,\infty )\)

Try it.

\(g(x)=\text{-}\sqrt[3]{x}\)

Try it.

\(f(x)=2\sqrt[3]{x}\)

Solution

ⓐ domain: \((\text{-}\infty ,\infty )\)
ⓑ

ⓒ \((\text{-}\infty ,\infty )\)

Try it.

\(f(x)=-2\sqrt[3]{x}\)

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Tani ti. Asnjë kalkulator nuk e zgjidh këtë, por pjesët e saj janë të llogaritura. Provo një më poshtë, ose shkruaj tënde.

⌨ Lloji ✎ E vizatova.

Problemi yt matematikor.

π Zgjidh

Praktiko (40)

Provo secilin në letër së pari. Zbulo përgjigjen për të kontrolluar; ato të verifikuara mund të hapen në zgjidhës për çdo hap.

1. Solve: \(1-2x\ge 0.\)

   Puno mbi të. Pjesmarrës

   Zbulo përgjigjen

   \((\text{-}\infty ,\frac{1}{2}]\)

2. For \(f(x)=3x-4,\) evaluate \(f(2),f(-1),f(0).\)

   Puno mbi të. Pjesmarrës

   Zbulo përgjigjen

   \(f(2)=2,f(-1)=-7,f(0)=-4\)

3. Graph \(f(x)=\sqrt{x}.\) State the domain and range of the function in interval notation.

   Puno mbi të. Pjesmarrës

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   \(\text{domain:}\ [0,\infty );\ \text{range:}\ [0,\infty )\)

4. For the function \(f(x)=\sqrt{2x-1},\) find ⓐ \(f(5)\) ⓑ \(f(-2).\)

   Puno mbi të. Pjesmarrës

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   ⓐ
   

   	\(f(x)=\sqrt{2x-1}\)
   To evaluate \(f(5),\) substitute 5 for \(x.\)	\(f(5)=\sqrt{2\cdot 5-1}\)
   Simplify.	\(f(5)=\sqrt{9}\)
   Take the square root.	\(f(5)=3\)

   ⓑ
   

   	\(\ f(x)=\sqrt{2x-1}\)
   To evaluate \(f(-2),\) substitute −2 for \(x.\)	\(\ f(-2)=\sqrt{2(-2)-1}\)
   Simplify.	\(\ f(-2)=\sqrt{-5}\)

   Since the square root of a negative number is not a real number, the function does not have a value at \(x=-2.\)

5. For the function \(f(x)=\sqrt{3x-2},\) find ⓐ \(f(6)\) ⓑ \(f(0).\)

   Puno mbi të. Pjesmarrës

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   ⓐ \(f(6)=4\) ⓑ no value at \(x=0\)

6. For the function \(g(x)=\sqrt{5x+5},\) find ⓐ \(g(4)\) ⓑ \(g(-3).\)

   Puno mbi të. Pjesmarrës

   Zbulo përgjigjen

   ⓐ \(g(4)=5\) ⓑ no value at \(f(-3)\)

7. For the function \(g(x)=\sqrt[3]{x-6},\) find ⓐ \(g(14)\) ⓑ \(g(-2).\)

   Puno mbi të. Pjesmarrës

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   ⓐ
   

   	\(\ g(x)=\sqrt[3]{x-6}\)
   To evaluate \(g(14),\) substitute 14 for \(x.\)	\(\ g(14)=\sqrt[3]{14-6}\)
   Simplify.	\(\ g(14)=\sqrt[3]{8}\)
   Take the cube root.	\(\ g(14)=2\)

   ⓑ
   

   	\(\ g(x)=\sqrt[3]{x-6}\)
   To evaluate \(g(-2),\) substitute −2 for \(x.\)	\(\ g(-2)=\sqrt[3]{-2-6}\)
   Simplify.	\(\ g(-2)=\sqrt[3]{-8}\)
   Take the cube root.	\(\ g(-2)=-2\)

8. For the function \(g(x)=\sqrt[3]{3x-4},\) find ⓐ \(g(4)\) ⓑ \(g(1).\)

   Puno mbi të. Pjesmarrës

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   ⓐ \(g(4)=2\) ⓑ \(g(1)=-1\)

9. For the function \(h(x)=\sqrt[3]{5x-2},\) find ⓐ \(h(2)\) ⓑ \(h(-5).\)

   Puno mbi të. Pjesmarrës

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   ⓐ \(h(2)=2\)
   ⓑ \(h(-5)=-3\)

10. For the function \(f(x)=\sqrt[4]{5x-4},\) find ⓐ \(f(4)\) ⓑ \(f(-12)\)

    Puno mbi të. Pjesmarrës

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    ⓐ
    

    	\(\ f(x)=\sqrt[4]{5x-4}\)
    To evaluate \(f(4),\) substitute 4 for \(x.\)	\(\ f(4)=\sqrt[4]{5\cdot 4-4}\)
    Simplify.	\(\ f(4)=\sqrt[4]{16}\)
    Take the fourth root.	\(\ f(4)=2\)

    ⓑ
    

    	\(\ f(x)=\sqrt[4]{5x-4}\)
    To evaluate \(f(-12),\) substitute −12 for \(x.\)	\(\ f(-12)=\sqrt[4]{5(-12)-4}\)
    Simplify.	\(\ f(-12)=\sqrt[4]{-64}\)

    Since the fourth root of a negative number is not a real number, the function does not have a value at \(x=-12.\)

11. For the function \(f(x)=\sqrt[4]{3x+4},\) find ⓐ \(f(4)\) ⓑ \(f(-1).\)

    Puno mbi të. Pjesmarrës

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    ⓐ \(f(4)=2\) ⓑ \(f(-1)=1\)

12. For the function \(g(x)=\sqrt[4]{5x+1},\) find ⓐ \(g(16)\) ⓑ \(g(3).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(g(16)=3\) ⓑ \(g(3)=2\)

13. Find the domain of the function, \(f(x)=\sqrt{3x-4}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    Since the function, \(f(x)=\sqrt{3x-4}\) has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0. We set the radicand to be greater than or equal to 0 and then solve to find the domain.

    	\(3x-4\ge 0\)
    Solve.	\(3x\ge 4\)
    	\(x\ge \frac{4}{3}\)

    The domain of \(f(x)=\sqrt{3x-4}\) is all values \(x\ge \frac{4}{3}\) and we write it in interval notation as \([\frac{4}{3},\infty ).\)

14. Find the domain of the function, \(f(x)=\sqrt{6x-5}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

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    \([\frac{5}{6},\infty )\)

15. Find the domain of the function, \(f(x)=\sqrt{4-5x}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

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    \((\text{-}\infty ,\frac{4}{5}]\)

16. Find the domain of the function, \(g(x)=\sqrt{\frac{6}{x-1}}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    Since the function, \(g(x)=\sqrt{\frac{6}{x-1}}\) has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0.

    The radicand cannot be zero since the numerator is not zero.

    For \(\frac{6}{x-1}\) to be greater than zero, the denominator must be positive since the numerator is positive. We know a positive divided by a positive is positive.

    We set \(x-1>0\) and solve.

    Solve.

    \(\begin{array}{lll}x-1 & > & 0 \\ x & > & 1\end{array}\)

    Also, since the radicand is a fraction, we must realize that the denominator cannot be zero.

    We solve \(x-1=0\) to find the value that must be eliminated from the domain.

    Solve.

    \(\begin{array}{lll}x-1 & = & 0 \\ x & = & 1\ \text{so}\ x\ne 1\ \text{in the domain.}\end{array}\)

    Putting this together we get the domain is \(x>1\) and we write it as \((1,\infty ).\)

17. Find the domain of the function, \(f(x)=\sqrt{\frac{4}{x+3}}.\) Write the domain in interval notation.

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    \((-3,\infty )\)

18. Find the domain of the function, \(h(x)=\sqrt{\frac{9}{x-5}}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

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    \((5,\infty )\)

19. Find the domain of the function, \(f(x)=\sqrt[3]{2{x}^{2}+3}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

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    Since the function, \(f(x)=\sqrt[3]{2{x}^{2}+3}\) has a radical with an index of 3, which is odd, we know the radicand can be any real number. This tells us the domain is any real number. In interval notation, we write \((\text{-}\infty ,\infty ).\)

    The domain of \(f(x)=\sqrt[3]{2{x}^{2}+3}\) is all real numbers and we write it in interval notation as \((\text{-}\infty ,\infty ).\)

20. Find the domain of the function, \(f(x)=\sqrt[3]{3{x}^{2}-1}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

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    \((\text{-}\infty ,\infty )\)

21. Find the domain of the function, \(g(x)=\sqrt[3]{5x-4}.\) Write the domain in interval notation.

    Puno mbi të. Pjesmarrës

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    \((\text{-}\infty ,\infty )\)

22. For the function \(f(x)=\sqrt{x+3},\)

    ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ Since the radical has index 2, we know the radicand must be greater than or equal to zero. If \(x+3\ge 0,\) then \(x\ge -3.\) This tells us the domain is all values \(x\ge -3\) and written in interval notation as \([-3,\infty ).\)

    ⓑ To graph the function, we choose points in the interval \([-3,\infty )\) that will also give us a radicand which will be easy to take the square root.

    ⓒ Looking at the graph, we see the y-values of the function are greater than or equal to zero. The range then is \([0,\infty ).\)

23. For the function \(f(x)=\sqrt{x+2},\) ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

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    ⓐ domain: \([-2,\infty )\)
    ⓑ
    
    ⓒ range: \([0,\infty )\)

24. For the function \(f(x)=\sqrt{x-2},\) ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

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    ⓐ domain: \([2,\infty )\)
    ⓑ
    
    ⓒ range: \([0,\infty )\)

25. For the function \(f(x)=\sqrt[3]{x},\) ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ Since the radical has index 3, we know the radicand can be any real number. This tells us the domain is all real numbers and written in interval notation as \((\text{-}\infty ,\infty )\)

    ⓑ To graph the function, we choose points in the interval \((\text{-}\infty ,\infty )\) that will also give us a radicand which will be easy to take the cube root.

    ⓒ Looking at the graph, we see the y-values of the function are all real numbers. The range then is \((\text{-}\infty ,\infty ).\)

26. For the function \(f(x)=\text{-}\sqrt[3]{x},\)

    ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ domain: \(\text{(}-\infty ,\infty )\)
    ⓑ
    
    ⓒ range: \(\text{(}-\infty ,\infty )\)

27. For the function \(f(x)=\sqrt[3]{x-2},\)

    ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ domain: \(\text{(}-\infty ,\infty )\)
    ⓑ
    
    ⓒ range: \(\text{(}-\infty ,\infty )\)

28. \(f(x)=\sqrt{4x-4},\) find ⓐ \(f(5)\) ⓑ \(f(0).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(f(5)=4\) ⓑ no value at \(x=0\)

29. \(f(x)=\sqrt{6x-5},\) find ⓐ \(f(5)\) ⓑ \(f(-1).\)

    Puno mbi të. Pjesmarrës

30. \(g(x)=\sqrt{6x+1},\) find ⓐ \(g(4)\) ⓑ \(g(8).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(g(4)=5\) ⓑ \(g(8)=7\)

31. \(g(x)=\sqrt{3x+1},\) find ⓐ \(g(8)\) ⓑ \(g(5).\)

    Puno mbi të. Pjesmarrës

32. \(F(x)=\sqrt{3-2x},\) find ⓐ \(F(1)\) ⓑ \(F(-11).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(F(1)=1\) ⓑ \(F(-11)=5\)

33. \(F(x)=\sqrt{8-4x},\) find ⓐ \(F(1)\) ⓑ \(F(-2).\)

    Puno mbi të. Pjesmarrës

34. \(G(x)=\sqrt{5x-1},\) find ⓐ \(G(5)\) ⓑ \(G(2).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(G(5)=2\sqrt{6}\) ⓑ \(G(2)=3\)

35. \(G(x)=\sqrt{4x+1},\) find ⓐ \(G(11)\) ⓑ \(G(2).\)

    Puno mbi të. Pjesmarrës

36. \(g(x)=\sqrt[3]{2x-4},\) find ⓐ \(g(6)\) ⓑ \(g(-2).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(g(6)=2\) ⓑ \(g(-2)=-2\)

37. \(g(x)=\sqrt[3]{7x-1},\) find ⓐ \(g(4)\) ⓑ \(g(-1).\)

    Puno mbi të. Pjesmarrës

38. \(h(x)=\sqrt[3]{{x}^{2}-4},\) find ⓐ \(h(-2)\) ⓑ \(h(6).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(h(-2)=0\) ⓑ \(h(6)=2\sqrt[3]{4}\)

39. \(h(x)=\sqrt[3]{{x}^{2}+4},\) find ⓐ \(h(-2)\) ⓑ \(h(6).\)

    Puno mbi të. Pjesmarrës

40. For the function \(f(x)=\sqrt[4]{2{x}^{3}},\) find ⓐ \(f(0)\) ⓑ \(f(2).\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(f(0)=0\) ⓑ \(f(2)=2\)

Mbaje punën tënde

Një llogari e lirë shtohet shënime në çdo mësim, një regjistrim të asaj që ju keni përfunduar, problemet tuaja të zgjidhura në një vend, dhe një mësues që ju mund të pyesni rreth kësaj faqeje. Matematika vetë është e hapur për të gjithë, të regjistruar apo jo.

Regjistrohu Hyr

Simbolet e përdorura këtu

Prek çdo simbol për përkufizimin e plotë, një fotografi dhe se çfarë do të thotë çdo shkronjë në të.

\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.\inftyinfinity
Not a number: "grows without bound" in limits and intervals.A \cup Bunion, intersection, difference
In either; in both; in A but not B.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\neqnot equal
The two sides are different.\supsupremum, infimum
Least upper bound, greatest lower bound.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Si: Use Radicals in Functions

1. Evaluate a radical function
2. Find the domain of a radical function
3. Graph radical functions
4. When
5. When
6. When the
7. When the

Pyetja që bëjnë njerëzit

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Pjesa e kësaj faqeje është adaptuar nga OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). E përmbledhur dhe ri-shkruar këtu; gabimet janë tona. Për shkak se burimi i tij është licencuar për përdorim jo komercial, ky mësim është i lirë, nuk shet asgjë dhe ndahet nën të njëjtën licencë.

Më shumë në Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value