9.1: The Square Root Function - Mathematics LibreTexts

Reflections

If we start with the basic equation \(y = \sqrt{x}\), then replace x with −x, then the graph of the resulting equation \(y = \sqrt{−x}\) is captured by reflecting the graph of \(y = \sqrt{x}\) (see Figure 1(c)) horizontally across the y-axis. The graph of \(y = \sqrt{−x}\) is shown in Figure 7(a).

Similarly, the graph of \(y = −\sqrt{x}\) would be a vertical reflection of the graph of \(y = \sqrt{x}\) across the x-axis, as shown in Figure 7(b).

More often than not, you will be asked to perform a reflection and a translation.

Example \(\PageIndex{6}\)

Sketch the graph of \(f(x) = \sqrt{4− x}\). Use the resulting graph to determine the domain and range of f.

First, rewrite the equation \(f(x) = \sqrt{4− x}\) as follows:

\(f(x) = \sqrt{−(x−4)}\)

Definition

Reflections First. It is usually more intuitive to perform reflections before translations.

With this thought in mind, we first sketch the graph of \(f(x) = \sqrt{−x}\), which is a reflection of the graph of \(f(x) = \sqrt{x}\)across the y-axis. This is shown in Figure 8(a).

Now, in \(f(x) = \sqrt{−x}\) replace x with x−4 to obtain \(f(x) = \sqrt{−(x−4)}\). This shifts the graph of \(f(x) = \sqrt{−x}\) four units to the right, as pictured in Figure 8(b).

To find the domain of the function \(f(x) = \sqrt{−(x−4)}\), or equivalently, \(f(x) = \sqrt{4−x}\), project each point on the graph of f onto the x-axis, as shown in Figure 9(a). Note that all real numbers less than or equal to 4 are shaded on the x-axis. Hence, the domain of f is

Domain = \((−\infty, 4]\) = {x: \(x \le 4\)}.

Similarly, to obtain the range of f, project each point on the graph of f onto they-axis, as shown in Figure 9(b). Note that all real numbers greater than or equal to zero are shaded on the y-axis. Hence, the range of f is

Range = \([0,\infty)\) = {x: \(x \ge 0\)}.

We can also find the domain of the function f by examining the equation \(f(x) = \sqrt{4−x}\). We cannot take the square root of a negative number, so the expression under the radical must be nonnegative (zero or positive). Consequently,

\(4 − x \ge 0\).

Solve this last inequality for x. First subtract 4 from both sides of the inequality, then multiply both sides of the resulting inequality by −1. Of course, multiplying by a negative number reverses the inequality symbol.

\(−x \ge −4\)

\(x \le 4\)

Thus, the domain of f is {x: \(x \le 4\)}. In interval notation, Domain = \((−\infty, 4]\). This agree nicely with the graphical result found above.

More often than not, it will take a combination of your graphing calculator and a little algebraic manipulation to determine the domain of a square root function.

Example \(\PageIndex{7}\)

Sketch the graph of \(f(x) = \sqrt{5−2x}\) Use the graph and an algebraic technique to determine the domain of the function.

Load the function into Y1 in the Y= menu of your calculator, as shown in Figure 10(a). Select 6:ZStandard from the ZOOM menu to produce the graph shown in Figure 10(b).

Look carefully at the graph in Figure 10(b) and note that it’s difficult to tell if the graph comes all the way down to “touch” the x-axis near \(x \approx 2.5\). However, our previous experience with the square root function makes us believe that this is just an artifact of insufficient resolution on the calculator that is preventing the graph from “touching” the x-axis at \(x \approx 2.5\).

An algebraic approach will settle the issue. We can determine the domain of f by examine the equation \(f(x) = \sqrt{5 − 2x}\). Consequently, We cannot take the square root of a negative number, so the expression under the radical must be nonnegative (zero or positive).

\(5 − 2x \ge 0\).

Solve this last inequality for x. First, subtract 5 from both sides of the inequality.

\(−2x \ge −5\).

Next, divide both sides of this last inequality by −2. Remember that we must reverse the inequality the moment we divide by a negative number.

\(\frac{−2x}{−2} \le \frac{−5}{−2}\).

\(x \le \frac{5}{2}\).

Thus, the domain of f is {x: \(x \le \frac{5}{2}\)}. In interval notation, Domain = \((−\infty, \frac{5}{2}]\). This agree nicely with the graphical result found above.

Further introspection reveals that this argument also settles the issue of whether or not the graph “touches” the x-axis at \(x= \frac{5}{2}\). If you remain unconvinced, then substitute \(x=\frac{5}{2}\) in \(f(x) = \sqrt{5−2x}\)to see

\(f(\frac{5}{2})= \sqrt{5−2(\frac{5}{2})} =\sqrt{0} = 0\).

Thus, the graph of f “touches” the x-axis at the point \((\frac{5}{2}, 0)\).

In Exercise 1-10, complete each of the following tasks:

1. Set up a coordinate system on a sheet of graph paper. Label and scale each axis.
2. Complete the table of points for the given function. Plot each of the points on your coordinate system, then use them to help draw the graph of the given function.
3. Use different colored pencils to project all points onto the x- and y-axes to determine the domain and range. Use interval notation to describe the do- main of the given function.

Exercise \(\PageIndex{1}\)

\(f(x) = −\sqrt{x}\)

x	0	1	4	9
f (x)

Answer

x	0	1	4	9
f (x)	0	−1	−2	−3

Plot the points in the table and use them to help draw the graph.

Project all points on the graph onto the x-axis to determine the domain: Domain = \([0, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \((−\infty, 0]\).

Exercise \(\PageIndex{2}\)

\(f(x) = \sqrt{−x}\)

x	0	−1	−4	−9
f (x)

Exercise \(\PageIndex{3}\)

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

x	−2	−1	2	7
f (x)

Answer

x	−2	−1	2	7
f (x)	0	1	2	3

Plot the points in the table and use them to help draw the graph.

Project all points on the graph onto the x-axis to determine the domain: Domain = \([−2, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \([0, \infty)\).

Exercise \(\PageIndex{4}\)

\(f(x)= \sqrt{5−x}\)

x	−4	1	4	5
f (x)

Exercise \(\PageIndex{5}\)

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

x	0	1	4	9
f (x)

Answer

x	0	1	4	9
f (x)	2	3	4	5

Plot the points in the table and use them to draw the graph of f.

Project all points on the graph onto the x-axis to determine the domain: Domain = \([0, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \([2, \infty)\).

Exercise \(\PageIndex{6}\)

\(f(x)=\sqrt{x}−1\)

x	0	1	4	9
f (x)

Exercise \(\PageIndex{7}\)

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

x	−3	−2	1	6
f (x)

Answer

x	−3	−2	1	6
f (x)	2	3	4	5

Plot the points in the table and use them to draw the graph of f.

Project all points on the graph onto the x-axis to determine the domain: Domain = \([−3, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \([2, \infty)\).

Exercise \(\PageIndex{8}\)

\(f(x)= \sqrt{x−1}+3\)

x	1	2	5	10
f (x)

Exercise \(\PageIndex{9}\)

\(f(x)= \sqrt{3−x}\)

x	−6	−1	2	3
f (x)

Answer

x	−6	−1	2	3
f (x)	3	2	1	0

Plot the points in the table and use them to draw the graph of f.

Project all points on the graph onto the x-axis to determine the domain: Domain = \((−\infty, 3]\). Project all points on the graph onto the y-axis to determine the range: Range = \([0, \infty)\).

Exercise \(\PageIndex{10}\)

\(f(x)=−\sqrt{x+3}\)

x	−3	−2	1	6
f (x)

In Exercises 11-20, perform each of the following tasks.

1. Set up a coordinate system on a sheet of graph paper. Label and scale each axis. Remember to draw all lines with a ruler.
2. Use geometric transformations to draw the graph of the given function on your coordinate system without the use of a graphing calculator. Note: You may check your solution with your calculator, but you should be able to produce the graph without the use of your calculator.
3. Use different colored pencils to project the points on the graph of the function onto the x- and y-axes. Use interval notation to describe the domain and range of the function.

Exercise \(\PageIndex{11}\)

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

Answer

First, plot the graph of \(y = \sqrt{x}\), as shown in (a). Then, add 3 to produce the equation \(y = \sqrt{x} + 3\). This will shift the graph of of \(y = \sqrt{x}\) upward 3 units, as shown in (b).

Project all points on the graph onto the x-axis to determine the domain: Domain = \([0, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \([3, \infty)\).

Exercise \(\PageIndex{12}\)

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

Exercise \(\PageIndex{13}\)

\(f(x)=\sqrt{x−2}\)

Answer

First, plot the graph of \(y = \sqrt{x}\), as shown in (a). Then, replace x with x − 2 to produce the equation \(y = \sqrt{x−2}\). This will shift the graph of \(y = \sqrt{x}\) to the right 2 units, as shown in (b).

Project all points on the graph onto the x-axis to determine the domain: Domain = \([2, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \([0, \infty)\).

Exercise \(\PageIndex{14}\)

\(f(x)=\sqrt{x}−2\)

Exercise \(\PageIndex{15}\)

\(f(x)= \sqrt{x+5}+1\)

Answer

First, plot the graph of \(y = \sqrt{x}\), as shown in (a). Then, replace x with x + 5 to produce the equation \(y = \sqrt{x+5}\). Then add 1 to produce the equation \(f(x)= \sqrt{x+5}+1\). This will shift the graph of \(y = \sqrt{x}\) to the left 5 units, then upwards 1 unit, as shown in (b).

Project all points on the graph onto the x-axis to determine the domain: Domain = \([−5, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \([1, \infty)\).

Exercise \(\PageIndex{16}\)

\(f(x)=\sqrt{x−2}−1\)

Exercise \(\PageIndex{17}\)

\(y = −\sqrt{x + 4}\)

Answer

First, plot the graph of \(y = \sqrt{x}\), as shown in (a). Then, negate to produce the \(y = −\sqrt{x}\). This will reflect the graph of \(y = \sqrt{x}\) across the x-axis as shown in (b). Finally, replace x with x + 4 to produce the equation \(y = −\sqrt{x + 4}\). This will shift the graph of \(y = −\sqrt{x}\) four units to the left, as shown in (c).

Project all points on the graph onto the x-axis to determine the domain: Domain = \([−4, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \((−\infty, 0]\).

Exercise \(\PageIndex{18}\)

\(f(x)=−\sqrt{x}+4\)

Exercise \(\PageIndex{19}\)

\(f(x)=−\sqrt{x}+3\)

Answer

First, plot the graph of \(y = \sqrt{x}\), as shown in (a). Then, negate to produce the \(y = −\sqrt{x}\). This will reflect the graph of \(y = \sqrt{x}\) across the x-axis as shown in (b). Finally, add 3 to produce the equation \(y=−\sqrt{x}+3\). This will shift the graph of \(y = −\sqrt{x}\) three units upward, as shown in (c).

Project all points on the graph onto the x-axis to determine the domain: Domain = \([0, \infty)\). Project all points on the graph onto the y-axis to determine the range: Range = \((−\infty, 3]\).

Exercise \(\PageIndex{20}\)

\(f(x)=−\sqrt{x+3}\)

Exercise \(\PageIndex{21}\)

To draw the graph of the function \(f(x) = \sqrt{3−x}\), perform each of the following steps in sequence without the aid of a calculator.

1. Set up a coordinate system and sketch the graph of \(y = \sqrt{x}\). Label the graph with its equation.
2. Set up a second coordinate system and sketch the graph of \(y = \sqrt{−x}\). Label the graph with its equation.
3. Set up a third coordinate system and sketch the graph of \(y =\sqrt{−(x − 3)}\). Label the graph with its equation. This is the graph of \(y =\sqrt{3−x}\). Use interval notation to state the domain and range of this function.

Answer

First, plot the graph of \(y = \sqrt{x}\), as shown in (a). Then, replace x with −x to produce the equation \(y = \sqrt{−x}\). This will reflect the graph of \(y = \sqrt{x}\) across the y-axis, as shown in (b). Finally, replace x with x − 3 to produce the equation \(y = \sqrt{−(x − 3)}\). This will shift the graph of \(y = \sqrt{−x}\) three units to the right, as shown in (c).

Project all points on the graph onto the x-axis to determine the domain: Domain = \((−\infty, 3]\). Project all points on the graph onto the y-axis to determine the range: Range = \([0, \infty)\).

Exercise \(\PageIndex{22}\)

To draw the graph of the function \(f(x) = \sqrt{−x−3}\), perform each of the following steps in sequence.

1. Set up a coordinate system and sketch the graph of \(y = \sqrt{x}\). Label the graph with its equation.
2. Set up a second coordinate system and sketch the graph of \(y = \sqrt{−x}\). Label the graph with its equation.
3. Set up a third coordinate system and sketch the graph of \(y =\sqrt{−(x + 3)}\). Label the graph with its equation. This is the graph of \(y =\sqrt{−x−3}\). Use interval notation to state the domain and range of this function.

Exercise \(\PageIndex{23}\)

To draw the graph of the function \(f(x) = \sqrt{−x−3}\), perform each of the following steps in sequence without the aid of a calculator.

1. Set up a coordinate system and sketch the graph of \(y = \sqrt{x}\). Label the graph with its equation.
2. Set up a second coordinate system and sketch the graph of \(y = \sqrt{−x}\). Label the graph with its equation.
3. Set up a third coordinate system and sketch the graph of \(y =\sqrt{−(x + 1)}\). Label the graph with its equation. This is the graph of \(y =\sqrt{−x−1}\). Use interval notation to state the domain and range of this function.

Answer

First, plot the graph of \(y = \sqrt{x}\), as shown in (a). Then, replace x with −x to produce the equation \(y = \sqrt{−x}\). This will reflect the graph of \(y = \sqrt{x}\) across the y-axis, as shown in (b). Finally, replace x with x + 1 to produce the equation \(y = \sqrt{−(x + 1)}\). This will shift the graph of \(y = \sqrt{−x}\) one unit to the left, as shown in (c).

Project all points on the graph onto the x-axis to determine the domain: Domain = \((−\infty, −1]\). Project all points on the graph onto the y-axis to determine the range: Range = \([0, \infty)\).

Exercise \(\PageIndex{24}\)

To draw the graph of the function \(f(x) = \sqrt{1−x}\), perform each of the following steps in sequence.

1. Set up a coordinate system and sketch the graph of \(y = \sqrt{x}\). Label the graph with its equation.
2. Set up a second coordinate system and sketch the graph of \(y = \sqrt{−x}\). Label the graph with its equation.
3. Set up a third coordinate system and sketch the graph of \(y =\sqrt{−(x−1)}\). Label the graph with its equation. This is the graph of \(y =\sqrt{1−x}\). Use interval notation to state the domain and range of this function.

In Exercises 25-28, perform each of the following tasks.

1. Draw the graph of the given function with your graphing calculator. Copy the image in your viewing window onto your homework paper. Label and scale each axis with xmin, xmax, ymin, and ymax. Label your graph with its equation. Use the graph to determine the domain of the function and describe the domain with interval notation.
2. Use a purely algebraic approach to determine the domain of the given function. Use interval notation to de- scribe your result. Does it agree with the graphical result from part 1?

Exercise \(\PageIndex{25}\)

\(f(x)= \sqrt{2x+7}\)

Answer

We use a graphing calculator to produce the following graph of \(f(x)= \sqrt{2x+7}\)

We estimate that the domain will consist of all real numbers to the right of approximately −3.5. To find an algebraic solution, note that you cannot take the square root of a negative number. Hence, the expression under the radical in \(f(x)= \sqrt{2x+7}\) must be greater than or equal to zero.

\(2x + 7 \ge 0\)

\(2x \ge −7\)

\(x \ge −\frac{7}{2}\)

Hence, the domain is \([−\frac{7}{2}, \infty)\).

Exercise \(\PageIndex{26}\)

\(f(x)= \sqrt{7−2x}\)

Exercise \(\PageIndex{27}\)

\(f(x)= \sqrt{12−4x}\)

Answer

We use a graphing calculator to produce the following graph of \(f(x)= \sqrt{12−4x}\).

We estimate that the domain will consist of all real numbers to the right of approximately 3. To find an algebraic solution, note that you cannot take the square root of a negative number. Hence, the expression under the radical in \(f(x)= \sqrt{12−4x}\) must be greater than or equal to zero.

\(12−4x \ge 0\)

\(−4x \ge −12\)

\(x \le 3\)

Hence, the domain is \((−\infty, 3]\).

Exercise \(\PageIndex{28}\)

\(f(x)= \sqrt{12+2x}\)

In Exercises 29-40, find the domain of the given function algebraically.

Exercise \(\PageIndex{29}\)

\(f(x)= \sqrt{2x+9}\)

Answer

The even root of a negative number is not defined as a real number. Thus, 2x + 9 must be greater than or equal to zero. Since \(2x + 9 \ge 0\) implies that \(x \ge −\frac{9}{2}\), the domain is the interval \([−\frac{9}{2},\infty)\).

Exercise \(\PageIndex{30}\)

\(f(x)=\sqrt{−3x+3}\)

Exercise \(\PageIndex{31}\)

\(f(x)=\sqrt{−8x−3}\)

Answer

The even root of a negative number is not defined as a real number. Thus, −8x−3 must be greater than or equal to zero. Since \(−8x−3 \ge 0\) implies that \(x \le −\frac{3}{8}\), the domain is the interval \((−\infty, −\frac{3}{8}]\).

Exercise \(\PageIndex{32}\)

\(f(x)=\sqrt{−3x+6}\)

Exercise \(\PageIndex{33}\)

\(f(x)=\sqrt{−6x−8}\)

Answer

The even root of a negative number is not defined as a real number. Thus, −6x−8 must be greater than or equal to zero. Since \(−6x−8 \ge 0\) implies that \(x \le −\frac{4}{3}\), the domain is the interval \((−\infty, \frac{4}{3}]\).

Exercise \(\PageIndex{34}\)

\(f(x)=\sqrt{8x−6}\)

Exercise \(\PageIndex{35}\)

\(f(x)=\sqrt{−7x+2}\)

Answer

The even root of a negative number is not defined as a real number. Thus, −7x+2 must be greater than or equal to zero. Since \(−7x+2 \ge 0\) implies that \(x \le \frac{2}{7}\), the domain is the interval \((−\infty, \frac{2}{7}]\).

Exercise \(\PageIndex{36}\)

\(f(x)=\sqrt{8x−3}\)

Exercise \(\PageIndex{37}\)

\(f(x)=\sqrt{6x+3}\)

Answer

The even root of a negative number is not defined as a real number. Thus, 6x+3 must be greater than or equal to zero. Since \(6x+3 \ge 0\) implies that \(x \ge −\frac{1}{2}\), the domain is the interval \([−\frac{1}{2}, \infty)\).

Exercise \(\PageIndex{38}\)

\(f(x)=\sqrt{x−5}\)

Exercise \(\PageIndex{39}\)

\(f(x)=\sqrt{−7x−8}\)

Answer

The even root of a negative number is not defined as a real number. Thus, −7x−8 must be greater than or equal to zero. Since \(−7x−8 \ge 0\) implies that \(x \le −\frac{8}{7}\), the domain is the interval \((−\infty, −\frac{8}{7}]\)

Exercise \(\PageIndex{40}\)

\(f(x)=\sqrt{7x+8}\)