# 5.7 Inverses and radical functions  (Page 4/7)

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## Solving applications of radical functions

Notice that the functions from previous examples were all polynomials, and their inverses were radical functions. If we want to find the inverse of a radical function , we will need to restrict the domain of the answer because the range of the original function is limited.

Given a radical function, find the inverse.

1. Determine the range of the original function.
2. Replace $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ then solve for $\text{\hspace{0.17em}}x.$
3. If necessary, restrict the domain of the inverse function to the range of the original function.

## Finding the inverse of a radical function

Restrict the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x-4}\text{\hspace{0.17em}}$ and then find the inverse.

Note that the original function has range $\text{\hspace{0.17em}}f\left(x\right)\ge 0.\text{\hspace{0.17em}}$ Replace $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ then solve for $\text{\hspace{0.17em}}x.$

Recall that the domain of this function must be limited to the range of the original function.

${f}^{-1}\left(x\right)={x}^{2}+4,x\ge 0$

Restrict the domain and then find the inverse of the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{2x+3}.$

${f}^{-1}\left(x\right)=\frac{{x}^{2}-3}{2},x\ge 0$

## Solving applications of radical functions

Radical functions are common in physical models, as we saw in the section opener. We now have enough tools to be able to solve the problem posed at the start of the section.

## Solving an application with a cubic function

A mound of gravel is in the shape of a cone with the height equal to twice the radius. The volume of the cone in terms of the radius is given by

$V=\frac{2}{3}\pi {r}^{3}$

Find the inverse of the function $\text{\hspace{0.17em}}V=\frac{2}{3}\pi {r}^{3}\text{\hspace{0.17em}}$ that determines the volume $\text{\hspace{0.17em}}V\text{\hspace{0.17em}}$ of a cone and is a function of the radius $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ Then use the inverse function to calculate the radius of such a mound of gravel measuring 100 cubic feet. Use $\text{\hspace{0.17em}}\pi =3.14.$

Start with the given function for $\text{\hspace{0.17em}}V.\text{\hspace{0.17em}}$ Notice that the meaningful domain for the function is $\text{\hspace{0.17em}}r>0\text{\hspace{0.17em}}$ since negative radii would not make sense in this context nor would a radius of 0. Also note the range of the function (hence, the domain of the inverse function) is $\text{\hspace{0.17em}}V>0.\text{\hspace{0.17em}}$ Solve for $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}V,\text{\hspace{0.17em}}$ using the method outlined previously. Note that in real-world applications, we do not swap the variables when finding inverses. Instead, we change which variable is considered to be the independent variable.

This is the result stated in the section opener. Now evaluate this for $\text{\hspace{0.17em}}V=100\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\pi =3.14.$

$\begin{array}{ccc}\hfill r& =& \sqrt{\frac{3V}{2\pi }}\hfill \\ & =& \sqrt{\frac{3\cdot 100}{2\cdot 3.14}}\hfill \\ & \approx & \sqrt{47.7707}\hfill \\ & \approx \hfill & 3.63\hfill \end{array}$

## Determining the domain of a radical function composed with other functions

When radical functions are composed with other functions, determining domain can become more complicated.

## Finding the domain of a radical function composed with a rational function

Find the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{\frac{\left(x+2\right)\left(x-3\right)}{\left(x-1\right)}}.$

Because a square root is only defined when the quantity under the radical is non-negative, we need to determine where $\text{\hspace{0.17em}}\frac{\left(x+2\right)\left(x-3\right)}{\left(x-1\right)}\ge 0.\text{\hspace{0.17em}}$ The output of a rational function can change signs (change from positive to negative or vice versa) at x -intercepts and at vertical asymptotes. For this equation, the graph could change signs at

To determine the intervals on which the rational expression is positive, we could test some values in the expression or sketch a graph. While both approaches work equally well, for this example we will use a graph as shown in [link] .

This function has two x -intercepts, both of which exhibit linear behavior near the x -intercepts. There is one vertical asymptote, corresponding to a linear factor; this behavior is similar to the basic reciprocal toolkit function, and there is no horizontal asymptote because the degree of the numerator is larger than the degree of the denominator. There is a y -intercept at $\text{\hspace{0.17em}}\left(0,\sqrt{6}\right).$

From the y -intercept and x -intercept at $\text{\hspace{0.17em}}x=-2,\text{\hspace{0.17em}}$ we can sketch the left side of the graph. From the behavior at the asymptote, we can sketch the right side of the graph.

From the graph, we can now tell on which intervals the outputs will be non-negative, so that we can be sure that the original function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ will be defined. $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ has domain $\text{\hspace{0.17em}}-2\le x<1\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}x\ge 3,\text{\hspace{0.17em}}$ or in interval notation, $\text{\hspace{0.17em}}\left[-2,1\right)\cup \left[3,\infty \right).$

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
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