Find F(X) and G(X) So the Function Can Be Expressed as Y = F(G(X)). Y = + 4
There are different means to Find the Range of a Function Algebraically. But before that, we take a brusk overview of the Range of a Role.
In the beginning chapter What is a Function? we have learned that a part is expressed as
y=f(x),
where ten is the input and y is the output.
For every input x (where the role f(x) is defined) there is a unique output.
The set of all outputs of a function is the Range of a Function.
The Range of a Function is the fix of all y values or outputs i.e., the set of all f(10) when it is defined.
We propose y'all read this article "9 Means to Find the Domain of a Function Algebraically" first. This volition help you to understand the concepts of finding the Range of a Function better.
In this article, you will larn
- v Steps to Observe the Range of a Part,
and in the end you will be able to
- Find the Range of 10 different types of functions
Table of Contents - What yous volition larn
Steps to Find the Range of a Part
Suppose nosotros take to find the range of the function f(x)=x+2.
We can find the range of a function by using the post-obit steps:
#1. First characterization the office equally y=f(x)
y=x+ii
#2. Express x as a office of y
Here x=y-2
#three. Find all possible values of y for which f(y) is divers
See that x=y-2 is defined for all real values of y.
#four. Element values of y by looking at the initial role f(x)
Our initial function y=x+2 is divers for all real values of 10 i.e., ten\epsilon \mathbb{R}.
So here we do not need to eliminate any value of y i.e., y\epsilon \mathbb{R}.
#5. Write the Range of the function f(x)
Therefore the Range of the function y=10+2 is {y\epsilon \mathbb{R}}.
Mayhap you are getting confused and don't understand all the steps now.
Merely believe me, you will get a clear concept in the adjacent examples.
How to Discover the Range of a Function Algebraically
In that location are unlike types of functions. Hither y'all will learn 10 ways to find the range for each type of function.
#i. Find the range of a Rational role
Example 1: Find the range
f(x)=\frac{x-2}{3-ten},x\neq3
Solution:
Pace 1: First we equate the function with y
y=\frac{x-2}{3-x}
Step 2: And so express x every bit a office of y
y=\frac{x-2}{three-x}
or, y(3-x)=ten-ii
or, 3y-xy=x-2
or, 10+xy=3y+ii
or, x(1+y)=3y+2
or, 10=\frac{3y+2}{y+1}
Step 3: Find possible values of y for which x=f(y) is defined
x=\frac{3y+2}{y+1} is defined when y+1 can not exist equal to 0,
i.e., y+ane\neq0
i.eastward., y\neq-1
i.due east., y\epsilon \mathbb{R}-{-1}
Step four: Eliminate the values of y
See that f(10)=\frac{x-two}{3-ten} is defined on \mathbb{R}-{3} and nosotros practice not need to eliminate any value of y from y\epsilon \mathbb{R}-{-i}.
Step 5: Write the Range
\therefore the range of f(ten)=\frac{ten-2}{3-x} is {x\epsilon \mathbb{R}:10\neq-i}.
Instance 2: Find the range
f(10)=\frac{3}{2-x^{two}}
Solution:
Step i:
y=\frac{3}{ii-10^{2}}
Step ii:
y=\frac{3}{2-x^{ii}}
or, 2y-xy^{2}=3
or, 2y-three=ten^{2y}
or, x^{2}=\frac{2y-3}{y}
Footstep 3:
The function x^{2}=\frac{2y-3}{y} is defined when y\neq 0 …(1)
Also since x^{2}\geq 0,
therefore
\frac{2y-3}{y}\geq 0
or, \frac{2y-3}{y}\times {\color{Magenta} \frac{y}{y}}\geq 0
or, \frac{y(2y-three)}{y^{two}}\geq 0
or, y(2y-three)\geq 0 (\because y^{2}\geq 0)
or, (y-0){\color{Magenta} 2}(y-\frac{iii}{{\colour{Magenta} 2}})
or, (y-0)(y-\frac{3}{two})\geq 0
Next we observe the values of y for which (y-0)(y-\frac{3}{2})\geq 0 i.e., y(2y-iii)\geq 0 is satisfied.
Now encounter the table:
| Value of y | Sign of (y-0) | Sign of (2y-3) | Sign of y(2y-3) | y(2y-iii)\geq 0 satisfied or non |
|---|---|---|---|---|
| y=-1<0 i.e., y\epsilon (-\infty,0) | -ve | -ve | +ve i.east., >0 | ✅ |
| y=0 | 0 | -ve | =0 | ✅ |
| y=1 i.e., y\epsilon (0,\frac{3}{ii}) | +ve | -ve | -ve i.due east., <0 | ❌ |
| y=\frac{three}{2} | +ve | 0 | =0 | ✅ |
| y=2>\frac{3}{2} i.e., y\epsilon (\frac{3}{two},\infty) | +ve | +ve | +ve i.e., >0 | ✅ |
Therefore from the higher up table and using (1) we get,
y\epsilon (-\infty,0)\cup [\frac{3}{2},\infty) (\because y\neq 0)
Step 4:
y=\frac{iii}{two-10^{ii}} is non a square office,
\therefore we do not need to eliminate any value of y except 0 considering if y be zero then the part y=\frac{3}{two-10^{2}} volition be undefined.
Footstep 5:
Therefore the range of the office f(x)=\frac{3}{2-x^{2}} is
(-\infty,0)\cup [\frac{3}{2},\infty).
Example 3: Find the range of a rational equation using changed
f(x)=\frac{2x-i}{ten+iv}
Solution:
#two. Find the range of a function with square root
Instance four: Notice the range
f(10)=\sqrt{4-10^{two}}
Solution:
Stride i: Offset we equate the office with y
y=\sqrt{4-x^{2}}
Step 2: So express x as a function of y
y=\sqrt{4-x^{2}}
or, y^{two}=4-x^{2}
or, ten^{2}=4-y^{2}
Footstep 3: Find possible values of y for which x=f(y) is divers
Since ten^{two}\geq 0,
\therefore 4-y^{ii}\geq 0
or, (2-y)(2+y)\geq 0
or, (y-two)(y+2)\leq 0
At present we find possible values for which (y-ii)(y+2)\leq 0
| Value of y | Sign of (y-2) | Sign of (y+2) | Sign of (y-two)(y+2) | (y-2)(y+2)\leq 0 is satisfied or not |
|---|---|---|---|---|
| y=-3<-2 i.e., y\epsilon (-\infty,-two) | -ve | -ve | +ve i.e., >0 | ❌ |
| y=-2 | -ve | 0 | =0 | ✅ |
| y=0 i.e., -two<y<2 i.e., y\epsilon (-2,2) | -ve | +ve | -ve i.e., <0 | ✅ |
| y=2 | 0 | +ve | =0 | ✅ |
| y=3>2 i.e., y\epsilon (2,\infty) | +ve | +ve | +ve i.east., >0 | ❌ |
i.e., y=-2, y\epsilon (-2,2) and y=2
i.e., y\epsilon [-ii,2]
Step iv: Eliminate the values of y
As y=\sqrt{4-ten^{2}}, a square root office,
so y tin not take any negative value i.e., y\geq 0
Therefore y\epsilon [0,2].
Step 5: Write the range
The range of the function f(x)=\sqrt{four-x^{2}} is [0,2] in interval notation.
Nosotros can also write the range of the function f(x)=\sqrt{4-10^{2}} as R(f)={10\epsilon \mathbb{R}:0\leq y \leq 2}
Example v: Discover the range of a function f(10) =\sqrt{x^{2}-four}.
Solution:
Step 1: First we equate the function with y
y=\sqrt{ten^{2}-4}
Step 2: And then limited x every bit a office of y
y=\sqrt{x^{2}-4}
or, y^{2}=x^{two}-iv
or, x^{two}=y^{2}+four
Footstep 3: Find possible values of y for which x=f(y) is divers
Since ten^{ii}\geq 0,
therefore y^{2}+four\geq 0
i.e., y^{ii}\geq -four
i.e., y\geq \sqrt{-4}
i.e, y\geq i\sqrt{2}, a complex number
\therefore y^{2}+iv\geq 0 for all y\epsilon \mathbb{R}
Stride iv: Eliminate the values of y
Sincey=\sqrt{x^{ii}-four} is a square root function,
therefore y can not take any negative value i.e.,y\geq 0
Step 5: Write the Range
The range of f(x) =\sqrt{x^{2}-four} is (0,\infty).
Case 6: Find the range for the square root office
f(ten)=iii-\sqrt{x}
Solution:
#3. Find the range of a role with a square root in the denominator
Case 7: Discover the range
f(x)=\frac{1}{\sqrt{x-iii}}
Solution:
Stride 1:
y=\frac{1}{\sqrt{10-three}}
Step 2:
y=\frac{1}{\sqrt{x-3}}
or, y=\frac{1}{\sqrt{x-3}}
or, y^{2}=\frac{1}{x-three}
or, y^{ii}(ten-iii)=ane
or, xy^{2}-3y^{2}=1
or, xy^{2}=1+3y^{2}
or, x=\frac{ane+3y^{2}}{y^{two}}
Step iii:
For x=\frac{1+3y^{2}}{y^{2}} to be defined,
y^{2}\neq 0
i.due east., y\neq 0
Footstep 4:
As f(x)=\frac{1}{\sqrt{x-iii}}, so y tin not be negative (-ve).
Stride 5:
The range of f(x)=\frac{i}{\sqrt{x-iii}} is (0,\infty).
Instance viii: Find the range
f(x)=\frac{1}{\sqrt{4-x^{two}}}
Solution:
Step 1:
y=\frac{1}{\sqrt{4-ten^{2}}}
Step ii:
y=\frac{i}{\sqrt{4-x^{2}}}
or, y=\frac{1}{\sqrt{four-ten^{2}}}
or, y^{2}=\frac{1}{iv-x^{2}}
or, 4y^{2}-x^{2}y^{2}=i
or, ten^{two}y^{2}=4y^{2}-i
or, x^{2}=\frac{4y^{2}-one}{y^{2}}
Stride 3:
For x^{ii}=\frac{4y^{two}-i}{y^{2}} to be defined, y tin can not be equal to zilch
i.e., y\neq 0
Likewise since x^{ii}\geq 0,
\therefore \frac{4y^{two}-1}{y^{2}}\geq 0
or, 4y^{2}-1\geq 0 (\considering y^{2}\geq 0)
or, (2y-i)(2y+i)\geq 0
or, four(y-\frac{1}{2})(y+\frac{1}{2})\geq 0
| Value of y | Sign of (2y-1) | Sign of (2y+1) | Sign of (2y-1)(2y+1) | (2y-1)(2y+one)\geq 0 is satisfied or non |
|---|---|---|---|---|
| y=-1<-\frac{one}{2} i.east., y\epsilon \left ( -\infty,-\frac{1}{2} \right ) | -ve | -ve | +ve i.e., >0 | ✅ |
| y=-\frac{one}{2} | -ve | 0 | 0 | ✅ |
| y=0 i.e., y\epsilon \left (-\frac{1}{ii},\frac{one}{ii} \correct ) | -ve | +ve | -ve i.e., <0 | ❌ |
| y=\frac{one}{2} | 0 | +ve | +ve i.e., >0 | ✅ |
| y=1>\frac{ane}{ii} i.e., y\epsilon \left (\frac{1}{ii},\infty \right ) | +ve | +ve | +ve i.e., >0 | ✅ |
The above table implies that
y\epsilon \left ( -\infty,-\frac{1}{two} \right )\loving cup \left (\frac{1}{two},\infty \right ) …..(1)
Step 4:
Since y=\frac{1}{\sqrt{4-ten^{two}}} is a square root office,
therefore y can not be negative (-ve).
i.e., y\geq 0 …..(2)
At present from (1) and (2), we get
y\epsilon ( \frac{one}{2},\infty )
Step v:
Therefore the range of the function f(ten)=\frac{1}{\sqrt{four-ten^{two}}} is [ \frac{1}{two},\infty )
Case 9: Notice the range of the part
f(x)=\sqrt{\frac{(10-3)(x+two)}{ten-1}}
Solution:
#4. Find the range of modulus function or accented value function
Example 10: Discover the range of the absolute value office
f(ten)=\left | ten \correct |
Solution:
Nosotros tin detect the range of the accented value part f(10)=\left | x \right | on a graph.
If we draw the graph then we get
Hither y'all tin see that the y value starts at y=0 and extended to infinity.
\therefore the range of the absolute value function f(x)=\left | x \right | is [0,\infty).
Example 11: Find the range of the absolute value function
f(x)=-\left | x-i \correct |
Solution:
The graph of f(x)=-\left | 10-1 \right | is
From the graph, it is articulate that the y value starts from y=0 and extended to -\infty.
Therefore the range of f(10)=-\left | x-i \correct | is (-\infty,0].
Shortcut Trick:
- If the sign before modulus is positive (+ve) i.e., of the class +\left | x-a \right |, then the range volition be [a,\infty),
- If the sign before modulus is negative (-ve) i.e., of the class -\left | x-a \right |, and so the range will exist (-\infty,a].
We can also find the range of the absolute value functions f(ten)=\left | x \right | and f(10)=-\left | x-one \right | using the to a higher place brusk cut trick:
The function f(ten)=\left | x \right | can be written equally f(x)=+\left | x-0 \right |
Now using trick 1 we can say, the range of f(ten)=\left | 10 \right | is [0,\infty)
Also using trick 2 we can say, the range of f(x)=-\left | x-1 \correct | is (-\infty,0].
Instance 12: Detect the range of the following absolute value functions
- f(x)=\left | x \correct |+6,
- f(ten)=\left | x+4 \right |
Solution:
#5. Discover the range of a Stride function
Example 13: Find the range of the step function f(x)=[ten],x\epsilon \mathbb{R}.
Solution:
The footstep office f(x)=[x],x\epsilon \mathbb{R} is expressed as
f(x)=0, 0\leq 10<1
=i, ane\leq 10<2
=2,2\leq x<3
………
=-1,-1\leq ten<0
=-ii,-2\leq x<-1
………
You lot can verify this consequence from the graph of f(x)=[x],x\epsilon \mathbb{R}
i.e., y\epsilon {…,-2,-1,0,1,2,…}
i.e., y\epsilon \mathbb{Z}, the set up of all integers.
\therefore the range of the step function f(x)=[x],10\epsilon \mathbb{R} is \mathbb{Z}, the set of all integers.
Example 14: Observe the range of the stride office f(x)=[x-3],ten\epsilon \mathbb{R}.
Solution:
Past using the definition of step part, nosotros tin can express f(x)=[x-3],x\epsilon \mathbb{R} as
f(x)=1,3\leq x<4
=2,4\leq x<5
=3,5\leq x<6
………
=0,2\leq x<3
=-i,i\leq 10<2
=-ii, 0\leq x<1
=-three, -i\leq x<0
………
You can verify this result from the graph of f(x)=[x-3],ten\epsilon \mathbb{R}
i.e., y\epsilon {…,-three,-ii,-one,0,one,2,3,…}
i.eastward., y\epsilon \mathbb{Z}, the set of all integers.
\therefore the range of the stride function f(x)=[x-3],x\epsilon \mathbb{R} is \mathbb{Z}, the set up of all integers.
Example 15: Find the range of the step function f(10)=\left [ \frac{i}{4x} \correct ],10\epsilon \mathbb{R}.
Solution:
#6. Notice the range of an Exponential part
Example 16: Detect the range of the exponential office f(x)=two^{x}.
Solution:
The graph of the function f(ten)=ii^{x} is
Hither y=0 is an asymptote of f(x)=two^{10} i.e., the graph is going very close and close to the y=0 straight line just it will never touch y=0.
Also, you can encounter on the graph that the function is extended to +\infty.
So we can say y>0.
\therefore the range of the exponential function f(ten)=2^{x} is (0,\infty).
Example 17: Observe the range of the exponential office
f(x)=-iii^{10+1}+2.
Solution:
The graph of the exponential role f(ten)=-3^{x+i}+ii is
From the graph of f(x)=-iii^{x+ane}+2 you can see that y=two is an asymptote of f(ten)=-3^{10+1}+2 i.e., on the graph f(x)=-3^{ten+1}+2 is going very close and close to y=2 towards -ve x-axis but it will never impact the straight line y=two and extended to -\infty towards +ve ten-axis.
i.e., y<ii
\therefore the range of the exponential function f(x)=-3^{x+one}+2 is (-\infty,2).
There is a shortcut trick to notice the range of any exponential function. This flim-flam volition help y'all notice the range of any exponential part in just 2 seconds.
Shortcut trick:
Let f(ten)=a\times b^{10-h}+k be an exponential part.
Then
- If a>0, then R(f)=(thou,\infty),
- If a<0, then R(f)=(-\infty,yard).
At present we try to find the range of the exponential functions f(x)=2^{x} and f(x)=-3^{x+1}+2 with the above shortcut trick:
We can write f(x)=ii^{x} as f(x)=i\times 2^{x}+0, one>0 and comparing this result with trick 1 we directly say
The range of f(x)=2^{10} is (0,\infty).
Too f(x)=-iii^{x+1}+2 can be written every bit f(x)=-1\times 3^{x+1}+2, -ane<0 and comparison with trick 2 nosotros go
The range of f(x)=-3^{x+i}+ii is (-\infty,2).
Example 18: Find the range of the exponential functions given below
f(x)=-2^{x+1}+three
Solution:
#7. Find the range of a Logarithmic part
The range of any logarithmic function is (-\infty,\infty).
We can verify this fact from the graph.
f(ten)=\log_{2}x^{3} is a logarithmic function and the graph of this part is
Here you lot tin meet that the y value starts from -\infty and extended to +\infty,
i.e., the range of f(x)=\log_{2}ten^{3} is (-\infty,\infty).
Example 19: Observe the range of the logarithmic role
f(10)=\log_{two}(x+4)+3
Solution:
#8. Find the range of a part relation of ordered pairs
A relation is the set of ordered pairs i.due east., the set of (x,y) where the set of all x values is called the domain and the fix of all y values is called the range of the relation.
In the previous chapter, nosotros take learned how to find the domain of a function using relation.
Now nosotros acquire how to find the range of a function using relation.
For that we have to recall 2 rules which are given below:
Rules:
- Before finding the range of a function get-go we bank check the given relation (i.e., the set of ordered pairs) is a part or not
- Observe all the y values and course a set. This set is the range of the relation.
Now see the examples given below to empathise this concept:
Instance 20: Find the range of the relation
{(one,3), (5,9), (8,23), (12,fourteen)}
Solution:
In the relation {(1,3), (five,9), (8,23), (12,14)}, the set of x coordinates is {ane, v, eight, 12} and the set of y coordinates is {3, 9, 14, 23}.
If nosotros describe the diagram of the given relation information technology will look like this
Here nosotros can clearly run into that each element of the set {1, 5, viii, 12} is related to a unique element of the set {3, nine, 14, 23}.
Therefore the given relation is a Function.
Besides, we know that the range of a role relation is the set up of y coordinates.
Therefore the range of the relation {(1,3), (5,nine), (8,23), (12,14)} is the set {3, nine, 14, 23}.
Example 21: Find the range of the set of ordered pairs
{(five,2), (7,6), (ix,4), (9,13), (12,nineteen)}.
Solution:
The diagram of the given relation is
Here we tin can see that element ix is related to two different elements and they are iv and thirteen i.e., nine is not related to a unique chemical element and this goes against the definition of the function.
Therefore the relation {(five,2), (7,vi), (9,4), (ix,13), (12,19)} is not a Function.
Instance 22: Decide the range of the relation described past the table
| ten | y |
|---|---|
| -1 | iii |
| iii | -2 |
| 3 | 2 |
| 4 | eight |
| 6 | -1 |
Solution:
#9. Notice the range of a Discrete part
A Discrete Office is a drove of some points on the Cartesian plane and the range of a detached function is the set of y-coordinates of the points.
Case 23: How exercise you find the range of the discrete function from the graph
Solution:
From the graph, nosotros tin see that there are 5 points on the detached office and they are A (2,2), B (4,iv), C (6,6), D (8,8), and E (x,10).
The set of the y-coordinates of the points A, B, C, D, and Eastward is {2,4,6, 8, 10}.
\therefore the range of the discrete function is {2,iv,half-dozen,8,10}.
Instance 24: Find the range of the discrete function from the graph
Solution:
The detached role is made of the v points A (-iii,two), B (-two,4), C (2,three), D (three,1), and E (5,five).
The gear up of the y coordinates of the detached part is {two,4,3,1,5} = {1,2,3,4,five}.
\therefore the range of the discrete function is {1,2,three,four,v}.
#ten. Find the range of a trigonometric part
| Trigonometric Function | Expresion | Range |
|---|---|---|
| Sine part | \sin x | [-ane,ane] |
| Cosine office | \cos ten | [-1,i] |
| Tangent function | \tan x | (-\infty,+\infty) |
| CSC function (Cosecant function) | \csc x | (-\infty,-i]\loving cup[i,+\infty) |
| Secant function | \sec ten | (-\infty,-ane]\cup[one,+\infty) |
| Cotangent function | \sec ten | (-\infty,+\infty) |
#11. Discover the range of an inverse trigonometric function
| Inverse trigonometric role | Expression | Range |
|---|---|---|
| Arc Sine role / Changed Sine office | \arcsin x or, \sin^{-1}x | [-\frac{\pi}{2},+\frac{\pi}{ii}] |
| Arc Cosine office / Inverse Cosine role | \arccos x or, \cos^{-i}ten | [0,\pi] |
| Arc Tangent function / Changed Tangent function | \arctan 10 or, \tan^{-one}x | (-\frac{\pi}{ii},+\frac{\pi}{2}) |
| Arc CSC function / Inverse CSC function | \textrm{arccsc}ten or, \csc^{-1}ten | [-\frac{\pi}{ii},0)\cup(0,\frac{\pi}{two}] |
| Arc Secant function / Inverse Secant role | \textrm{arcsec}10 or, \sec^{-1}x | [0,\frac{\pi}{2})\cup(\frac{\pi}{two},\pi] |
| Arc Cotangent function / Inverse Cotangent role | \textrm{arccot}x or, \cot^{-1}10 | (0,\pi) |
#12. Discover the range of a hyperbolic office
| Hyperbolic function | Expression | Range |
|---|---|---|
| Hyperbolic Sine function | \sinh x=\frac{east^{x}-e^{-x}}{2} | (-\infty,+\infty) |
| Hyperbolic Cosine function | \cosh x=\frac{e^{x}+e^{-ten}}{2} | [1,\infty) |
| Hyperbolic Tangent role | \tanh x=\frac{east^{x}-e^{-x}}{e^{x}+e^{-ten}} | (-i,+1) |
| Hyperbolic CSC part | csch x=\frac{2}{e^{x}-e^{-x}} | (-\infty,0)\cup(0,\infty) |
| Hyperbolic Secant function | sech 10=\frac{2}{e^{ten}+e^{-ten}} | (0,1) |
| Hyperbolic Cotangent function | \tanh 10=\frac{e^{x}+e^{-ten}}{e^{x}-due east^{-x}} | (-\infty,-1)\cup(one,\infty) |
#13. Find the range of an changed hyperbolic function
| Inverse hyperbolic function | Expression | Range |
|---|---|---|
| Inverse hyperbolic sine function | \sinh^{-1}x=\ln(10+\sqrt{10^{2}+1}) | (-\infty,\infty) |
| Inverse hyperbolic cosine role | \cosh^{-1}x=\ln(ten+\sqrt{x^{ii}-1}) | [0,\infty) |
| Inverse hyperbolic tangent function | \tanh^{-1}x=\frac{one}{2}\ln\left (\frac{i+x}{1-x}\right ) | (-\infty,\infty) |
| Inverse hyperbolic CSC role | csch^{-1}x=\ln \left ( \frac{i+\sqrt{1+10^{ii}}}{x} \correct ) | (-\infty,0)\loving cup(0,\infty) |
| Inverse hyperbolic Secant office | sech^{-1}ten=\ln \left ( \frac{1+\sqrt{1-x^{2}}}{10} \right ) | [0,\infty) |
| Inverse hyperbolic Cotangent office | coth^{-1}x=\frac{one}{2}\ln\left (\frac{x+1}{x-1}\right ) | (-\infty,0)\cup(0,\infty) |
#xiv. Discover the range of a piecewise function
Example 25: Discover the range of the piecewise office
Solution:
The piecewise function consists of two function:
- f(x)=x-3 when x\leq -ane,
- f(x)=x+ane when x>1.
If nosotros plot these 2 functions on the graph then nosotros get,
This is the graph of the piecewise function.
From the graph, we can encounter that
- the range of the part f(x)=ten-3 is (-\infty,-2] when 10\leq -1,
- the range of the office f(x)=x+one is (2,\infty) when 10>1,
Therefore from the above results we tin say that
The range of the piecewise function f(ten) is
(-\infty,-ii]\cup (2,\infty).
Instance 26: Observe the range of a piecewise function given beneath
Solution:
If you notice the piecewise function so you lot tin see in that location are functions:
- f(ten)=x divers when x\leq -1,
- f(x)=2 defined when -1<x<i),
- f(10)=\sqrt{x} defined when x\geq 1.
Now if we draw the graph of these 3 functions we go,
This is the graph of the piecewise function.
Here you lot can come across that
The function f(ten)=10 starts y=-1 and extended to -\infty when 10\leq -ane.
So the range of the office f(x)=x,10\leq -1 is (-\infty,-1]……..(1)
The functional value of the part f(10)=ii, -1<10<one is 2.
The range of the part f(ten)=x is {ii}……..(2)
The function f(x)=\sqrt{x} starts at y=1 and extended to \infty when x\geq one.
The range of the part f(x)=\sqrt{x} is [1,\infty) when 10\geq ane……..(iii)
From (1), (ii), and (3), we get,
the range of the piecewise role is
(-\infty,-1]\cup {2}\cup [1,\infty)
= (-\infty,-1]\cup [1,\infty)
#15. Observe the range of a composite function
Case 27: Let f(x)=2x-six and g(ten)=\sqrt{x} exist two functions.
Find the range of the following blended functions:
(a) f\circ thou(x)
(b) thou\circ f(x)
Solution of (a)
Starting time we demand to discover the function chiliad\circ f(x).
We know that,
f\circ grand(x)
=f(g(x))
=f(\sqrt{ten}) (\because g(x)=\sqrt{x})
=ii\sqrt{ten}-6
Now run into that 2\sqrt{x}-half-dozen is a part with a foursquare root and at the beginning of this article, we already learned how to notice the range of a function with a square root.
Following these steps, we can go,
the range of the composite function f of chiliad is
R(f\circ g)=[-6,\infty).
Solution of (b):
g\circ f(x)
=g(f(10))
=1000(2x-vi) (\considering f(x)=2x-6)
=\sqrt{2x-6}, a office with a square root
Using the previous method we get,
the range of the composite part g\circ f(x) is
R(grand\circ f(10))=[0,\infty)
Example 28: Let f(10)=3x-12 and g(10)=\sqrt{x} exist two functions.
Detect the range of the following composite functions
- f\circ one thousand(10),
- yard\circ f(x)
Solution:
Anil Kumar (Duration: 3 minutes 35 seconds)
Also read:
- How to Find the Domain of a Office Algebraically – Best 9 Means
- 3 ways to detect the zeros of a function
- How to find the zeros of a quadratic function?
- thirteen ways to find the limit of a function
- How to utilise the Squeeze theorem to discover a limit?
schreiberoblem1944.blogspot.com
Source: https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/
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