High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Radical Functions

Rational FunctionsPrevious Section
Piecewise FunctionsNext Section
Popular Courses
Find My Course
0:00 / 0:00

Radical Functions

A radical function is defined as:
y=  nf(x)\boxed{y=~~^n\sqrt {f(x)}}
where f(x)f(x) is a function and n∈Wn\in\mathbb{W}.

Square-Root Functions

y=xy=\sqrt{x}

xy00114293164\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}

xx-Intercept: (0,0)(0,0)
yy-Intercept: (0,0)(0,0)
Domain: [0,∞)[0,\infin)
Range: [0,∞)[0,\infin)
0:00 / 0:00

Solving Radical Equations

A radical equation is
f(x)=g(x)\boxed{\sqrt{f(x)}=g(x)}
where f(x) & g(x) f(x)~\&~g(x)~ are continuous polynomial functions.

Solutions to Radical Equations

Solve 2x+1=32\sqrt{x}+1=3. State the restrictions for 'x'.


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

Restrictions: x≥0x\geq0

2x+1=32\sqrt{x}+1=3
  ⟹  2x=2\implies 2\sqrt{x}=2
  ⟹  x=1\implies \sqrt{x}=1
  ⟹  x=1\implies x=1

Graph f(x) f(x)~ and g(x)g(x):


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

Let's verify the answer:
2x+1=321+1=33=3✓\begin{array}{cccc} 2\sqrt{x}+1&=&3\\\\ 2\sqrt{1}+1&=&3\\\\ 3&=&3&\checkmark \end{array}
Therefore, (1, 3) is the solution.
PAGE BREAK

Algebraic Solutions to Radical Equations

Solve 5x+6−2=x\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≥05x+6\ge0 which means that our domain can be given by x≥−65x\ge-\frac65, or [−65,∞)[-\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,∞)[-2,\infin).

5x+6−2=x(5x+6)2=(x+2)25x+6=x2+4x+40=x2−x−20=(x−2)(x+1)∴  x=−1,2\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:
5(−1)+6−2=−1−5+6−2=−11−2=−1−1==−1✓                     5(2)+6−2=210+6−2=216−2=22=2✓\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}

0:00 / 0:00

Example: Solving Radical Equations

Solve graphically & algebraically, stating the restrictions on xx:
x−4+x=4\sqrt{x-4}+x=4

Graphically:

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


The solution is (4, 4)

Restrictions on 'x': x≥4x\geq4



PAGE BREAK

Algebraically:

x−4+x=4\sqrt{x-4}+x=4
(x−4)2=(−x+4)2x−4=x2−8x+160=x2−9x+200=(x−5)(x−4)∴  x=4,5\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:
x−4+x=44−4+4=44=4✓                  5−4+5=45−4+5=46≠4   Extraneous Solution\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:
x+7+2=3−x\sqrt{x+7}+2=\sqrt{3-x}

Extra Practice
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Function composition
Consider the functions f(x)=x1/23−xf(x)=\frac{x^{1/2}}{\sqrt{3-\sqrt{x}}} and g(x)=(6−x)2g(x)=(6-x)^2. Find (f∘g)(x)(f\circ g)(x) and the domain.
Absolute Value Functions: Function Properties
If f(x)=∣x−2∣, g(x)=3−x+1f\left(x\right)=\left|x-2\right|,\ g\left(x\right)=3-\sqrt{x+1}, which of the following is true about h(x)=g(f(x))h\left(x\right)=g\left(f\left(x\right)\right)?
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Radical Functions
State the domain and range of the function f(x)=1x2−4f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}}.
Consider the functions f(x)=x25−x2f\left(x\right)=\dfrac{x^2}{\sqrt{5-x^2}} and g(x)=2−xg\left(x\right)=\sqrt{2-x}
a. Find (f∘g)(x)(f \circ g)(x)
b. Find the domain of (f∘g)(x)(f \circ g)(x) . Enter in interval notation.
Radical Functions
Find the domain of f(x)=1x2−4−3f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}-3}
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Radical Functions
Solve for xx:
3x3−3x2+1=x^3\sqrt{x^{3}-3x^2}+1=x
Solving Radical Equations
Solve for a:
2a+4−3=3a−22\sqrt{a+4}-3=\sqrt{3a-2}
Solving Radical Equations
Solve for xx:
2x−1=x\sqrt{2x-1}=\sqrt{x}
Radical Functions
Graph the function y=−34(x−2)+3y=-3\sqrt{4(x-2)}+3.
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions:
x−6+3=x+9\sqrt{x-6}+3=\sqrt{x+9}
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions:
x2−2=x+4\sqrt{x^2-2}=\sqrt{x+4}
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions on xx:
10+10x−1=13\begin{array}{rcl} 10+\sqrt{10x-1}&=&13 \end{array}
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions on xx:
x−3=3x−4\begin{array}{rcl} x-3&=&3\sqrt{x-4} \end{array}
Transformations of Radical Functions
Practice: Transformations of Radical Functions
Which of the following is the graph of the function y=−−3x+6−2y=-\sqrt{-3x+6}-2?
Transformations of Radical Functions
Practice: Transformations of Radical Functions
Which of the following is the graph of the function y=4x−1y=\sqrt{4x}-1?
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Absolute Value Functions: Function Properties
If f(x)=∣x−2∣, g(x)=3−x+1f\left(x\right)=\left|x-2\right|,\ g\left(x\right)=3-\sqrt{x+1}, which of the following is true about h(x)=g(f(x))h\left(x\right)=g\left(f\left(x\right)\right)?
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Consider the functions f(x)=x25−x2f\left(x\right)=\dfrac{x^2}{\sqrt{5-x^2}} and g(x)=2−xg\left(x\right)=\sqrt{2-x}
a. Find (f∘g)(x)(f \circ g)(x)
b. Find the domain of (f∘g)(x)(f \circ g)(x) . Enter in interval notation.
Radical Functions
Find the domain of f(x)=1x2−4−3f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}-3}
Radical Functions
State the domain and range of the function f(x)=1x2−4f\left(x\right)=\dfrac{1}{\sqrt{x^2-4}}.
Function composition
Consider the functions f(x)=x1/23−xf(x)=\frac{x^{1/2}}{\sqrt{3-\sqrt{x}}} and g(x)=(6−x)2g(x)=(6-x)^2. Find (f∘g)(x)(f\circ g)(x) and the domain.
Radical Functions
Given the function g(x)=3+5+7xg\left(x\right)=3+\sqrt{5+7x}
Fill in the blanks using the following options:
{x : x<75},  {x : x≥75} ,  {x : x≤75} , {x : x>75} \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\}\