Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Radical Functions

Radical Functions

A radical function is defined as:

\boxed{y=~~^n\sqrt {f(x)}}

where $f(x)$ is a function and $n\in\mathbb{W}$ .

Square-Root Functions

$y=\sqrt{x}$

Image illustrating Radical Functions $\begin{array}{|c|c|}\hline x&y\\\hline 0&0\\\hline 1&1\\\hline 4&2\\\hline 9&3\\\hline 16&4\\\hline \end{array}$

$x$ -Intercept: $(0,0)$

$y$ -Intercept: $(0,0)$

Domain: $[0,\infin)$

Range: $[0,\infin)$

Solving Radical Equations

A radical equation is

\boxed{\sqrt{f(x)}=g(x)}

where $f(x)~\&~g(x)~$ are continuous polynomial functions.

Solutions to Radical Equations

Solve $2\sqrt{x}+1=3$ . State the restrictions for 'x'.

Let $f(x)=2\sqrt{x}+1$ and let $g(x)=3$ .

Restrictions: $x\geq0$

$2\sqrt{x}+1=3$

$\implies 2\sqrt{x}=2$

$\implies \sqrt{x}=1$

$\implies x=1$

Graph $f(x)~$ and $g(x)$ :

Image illustrating Solving Radical Equations

$f(x)~\&~g(x)~$ intersect at (1, 3).

Let's verify the answer:

\begin{array}{cccc} 2\sqrt{x}+1&=&3\\\\ 2\sqrt{1}+1&=&3\\\\ 3&=&3&\checkmark \end{array}

Therefore, (1, 3) is the solution.

Algebraic Solutions to Radical Equations

Solve $\sqrt{5x+6}-2=x$ . State the domain and range

Isolate for 'x'

Restriction: Since the argument of a square root must be 0 or larger, $5x+6\ge0$ which means that our domain can be given by $x\ge-\frac65$ , or $[-\frac65,\infin)$ in interval notation. The range can be found be recognizing that the square root has an output of 0 or higher, and so subtracting 2 from this will give us a possible range of $[-2,\infin)$ .

\begin{array}{ccc} \sqrt{5x+6}-2&=&x\\\\ (\sqrt{5x+6})^2&=&(x+2)^2\\\\ 5x+6&=&x^2+4x+4\\\\ 0&=&x^2-x-2\\\\ 0&=&(x-2)(x+1)\\\\ \therefore~~x&=&-1, 2 \end{array}

Verify:

\begin{array}{rccc} \sqrt{5(-1)+6}-2&=&-1\\\\ \sqrt{-5+6}-2&=&-1\\\\ \sqrt{1}-2&=&-1\\\\ -1&=&=-1&\checkmark \end{array} ~~~~~~~~~~~~~~~~~~~~~ \begin{array}{rccc} \sqrt{5(2)+6}-2&=&2\\\\ \sqrt{10+6}-2&=&2\\\\ \sqrt{16}-2&=&2\\\\ 2&=&2&\checkmark \end{array}

Example: Solving Radical Equations

Solve graphically & algebraically, stating the restrictions on $x$ :

\sqrt{x-4}+x=4

Graphically:

Let $f(x)=\sqrt{x-4}+x$ and $g(x)=4$ . Then,

The solution is (4, 4)

Restrictions on 'x': $x\geq4$

Algebraically:

$\sqrt{x-4}+x=4$

\begin{array}{rcl} (\sqrt{x-4})^2&=&(-x+4)^2\\\\ x-4&=&x^2-8x+16\\\\ 0&=&x^2-9x+20\\\\ 0&=&(x-5)(x-4)\\\\ \therefore~~x&=&4, 5 \end{array}

Verify:

\begin{array}{rcl} \sqrt{x-4}+x&=&4\\\\ \sqrt{4-4}+4&=&4\\\\ 4&=&4&\checkmark \end{array} ~~~~~~~~~~~~~~~~~~ \begin{array}{rcl} \sqrt{5-4}+5&=&4\\\\ \sqrt{5-4}+5&=&4\\\\ 6&\neq&4&&~~~\footnotesize{\text{Extraneous Solution}} \end{array}

Therefore, only x = 4 is a solution.

Solve algebraically, stating any restrictions:

\sqrt{x+7}+2=\sqrt{3-x}

A) $x=-6$ only

B) $x=2$ only

C) $x=-6$ and $x=2$ only

D) $x=-2$ only

E) $x=-6$ and $x=-2$ only

I don't know

Extra Practice

Radical Functions

Given the function $g\left(x\right)=3+\sqrt{5+7x}$

Fill in the blanks using the following options:

$\left\{x\ :\ x<\frac{7}{5}\right\},\ \ \left\{x\ :\ x\ge\frac{7}{5}\right\}\ ,\ \ \left\{x\ :\ x\le\frac{7}{5}\right\}\ ,\ \left\{x\ :\ x>\frac{7}{5}\right\}\$

Function composition

Consider the functions $f(x)=\frac{x^{1/2}}{\sqrt{3-\sqrt{x}}}$ and $g(x)=(6-x)^2$ . Find $(f\circ g)(x)$ and the domain.

Radical Functions

State the domain and range of the function $f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}}$ .

Radical Functions

Find the domain of $f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}-3}$

Consider the functions $f\left(x\right)=\dfrac{x^2}{\sqrt{5-x^2}}$ and $g\left(x\right)=\sqrt{2-x}$

a. Find $(f \circ g)(x)$

b. Find the domain of $(f \circ g)(x)$ . Enter in interval notation.

Domain & Range

Practice: Domain & Range

If $f\left(x\right)=2-\sqrt{x}$ and $g\left(x\right)=x^2-9$ , find the domain and range of $f\left(g\left(x\right)\right)$ .

Absolute Value Functions: Function Properties

If $f\left(x\right)=\left|x-2\right|,\ g\left(x\right)=3-\sqrt{x+1}$ , which of the following is true about $h\left(x\right)=g\left(f\left(x\right)\right)$ ?

Domain & Range

Practice: Domain & Range

If $f\left(x\right)=2-\sqrt{x}$ and $g\left(x\right)=x^2-9$ , find the domain and range of $f\left(g\left(x\right)\right)$ .

Domain & Range

Practice: Domain & Range

If $f\left(x\right)=2-\sqrt{x}$ and $g\left(x\right)=x^2-9$ , find the domain and range of $f\left(g\left(x\right)\right)$ .

Transformations of Radical Functions

Practice: Transformations of Radical Functions

Which of the following is the graph of the function $y=\sqrt{4x}-1$ ?

Transformations of Radical Functions

Practice: Transformations of Radical Functions

Which of the following is the graph of the function $y=-\sqrt{-3x+6}-2$ ?

Solving Radical Equations

Practice: Solving Radical Equations

Solve algebraically, stating any restrictions on $x$ :

\begin{array}{rcl} x-3&=&3\sqrt{x-4} \end{array}

Solving Radical Equations

Practice: Solving Radical Equations

Solve algebraically, stating any restrictions on $x$ :

\begin{array}{rcl} 10+\sqrt{10x-1}&=&13 \end{array}

Solving Radical Equations

Practice: Solving Radical Equations

Solve algebraically, stating any restrictions:

\sqrt{x^2-2}=\sqrt{x+4}

Solving Radical Equations

Practice: Solving Radical Equations

Solve algebraically, stating any restrictions:

\sqrt{x-6}+3=\sqrt{x+9}

Radical Functions

Graph the function $y=-3\sqrt{4(x-2)}+3$ .

Solving Radical Equations

Solve for $x$ :

\sqrt{2x-1}=\sqrt{x}

Solving Radical Equations

Solve for a:

$2\sqrt{a+4}-3=\sqrt{3a-2}$

Radical Functions

Solve for $x$ :

$^3\sqrt{x^{3}-3x^2}+1=x$

Domain & Range

Practice: Domain & Range

If $f\left(x\right)=2-\sqrt{x}$ and $g\left(x\right)=x^2-9$ , find the domain and range of $f\left(g\left(x\right)\right)$ .

Radical Functions

Find the domain of $f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}-3}$

Consider the functions $f\left(x\right)=\dfrac{x^2}{\sqrt{5-x^2}}$ and $g\left(x\right)=\sqrt{2-x}$

a. Find $(f \circ g)(x)$

b. Find the domain of $(f \circ g)(x)$ . Enter in interval notation.

Radical Functions

State the domain and range of the function $f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}}$ .

Domain & Range

Practice: Domain & Range

If $f\left(x\right)=2-\sqrt{x}$ and $g\left(x\right)=x^2-9$ , find the domain and range of $f\left(g\left(x\right)\right)$ .

Absolute Value Functions: Function Properties

If $f\left(x\right)=\left|x-2\right|,\ g\left(x\right)=3-\sqrt{x+1}$ , which of the following is true about $h\left(x\right)=g\left(f\left(x\right)\right)$ ?

Function composition

Consider the functions $f(x)=\frac{x^{1/2}}{\sqrt{3-\sqrt{x}}}$ and $g(x)=(6-x)^2$ . Find $(f\circ g)(x)$ and the domain.

Domain & Range

Practice: Domain & Range

If $f\left(x\right)=2-\sqrt{x}$ and $g\left(x\right)=x^2-9$ , find the domain and range of $f\left(g\left(x\right)\right)$ .

Radical Functions

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