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